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level-4-ball-deterministic-collisions.html
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level-4-ball-deterministic-collisions.html
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<h2 class="mume-header" id="goal">goal</h2>
<ul>
<li>Simulate the motion of multiple balls.</li>
<li>There is no gravity, our simulation may be interpreted as a top-down view on a billiard table.</li>
<li>Collisions can happen with all the walls, and between each of the balls.</li>
<li>All balls, except one, are initially stationary.</li>
</ul>
<br>
<h2 class="mume-header" id="equations">equations</h2>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— constants ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- constants ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord text"><span class="mord">——— constants ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mspace width="5em"></mspace><mi>w</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mtext>const.</mtext><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mspace width="1.5555555555555556em"></mspace><mtext>(width of the scene)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>h</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mtext>const.</mtext><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mspace width="1.5555555555555556em"></mspace><mtext>(height of the scene)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\text{\hspace{50pt}}
w &= \text{const.}\text{\small\color{Gray}\hspace{14pt}(width of the scene)}\\
h &= \text{const.}\text{\small\color{Gray}\hspace{14pt}(height of the scene)}\\[8pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8000000000000007em;vertical-align:-1.6500000000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span style="top:-4.3100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mspace" style="margin-right:5em;"></span></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span><span style="top:-2.8099999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">h</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6500000000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1500000000000004em;"><span style="top:-4.3100000000000005em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord">const.</span></span><span class="mord text"><span class="mspace sizing reset-size6 size5" style="color:Gray;margin-right:1.5555555555555556em;"></span><span class="mord sizing reset-size6 size5" style="color:Gray;">(width of the scene)</span></span></span></span><span style="top:-2.8099999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord">const.</span></span><span class="mord text"><span class="mspace sizing reset-size6 size5" style="color:Gray;margin-right:1.5555555555555556em;"></span><span class="mord sizing reset-size6 size5" style="color:Gray;">(height of the scene)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6500000000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— time step ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- time step ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8623000000000001em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">——— time step ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (for every ball i ...)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (for every ball i ...)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (for every ball i ...)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (new position, using the new velocity)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (new position, using the new velocity)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (new position, using the new velocity)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right left right left" columnspacing="0em 1em 0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>x</mi><mi>i</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>d</mi><msub><mi>x</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>←</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>d</mi><msub><mi>x</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>d</mi><mi>t</mi><mo>⋅</mo><msub><mi>v</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>y</mi><mi>i</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo>+</mo><mi>d</mi><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>←</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>d</mi><msub><mi>y</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>d</mi><mi>t</mi><mo>⋅</mo><msub><mi>v</mi><mrow><mi>y</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
x'_i &= x_i + dx_i & &\leftarrow & dx_i &= dt \cdot v_{x,i}\\[8pt]
y'_i &= y_i + dy_i & &\leftarrow & dy_i &= dt \cdot v_{y,i}\\[8pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.600000000000001em;vertical-align:-2.050000000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.71em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.71em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.55em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-2.249999999999999em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.71em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">←</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.71em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5500000000000003em;"><span style="top:-4.71em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.050000000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— ball-ball collisions ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- ball-ball collisions ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">——— ball-ball collisions ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (for every pair of balls i and j ...)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (for every pair of balls i and j ...)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (for every pair of balls i and j ...)</span></span></span></span></span></span></p>
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mi>i</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mspace width="1.6em"></mspace><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mi>j</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>j</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>d</mi><mo>⃗</mo></mover><mo>=</mo><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mrow><mspace width="0.05em"></mspace><mi>j</mi></mrow></msub><mo>−</mo><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mrow><mspace width="0.05em"></mspace><mi>i</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mtext>collision condition:</mtext><mspace width="1.2000000000000002em"></mspace></mrow><mi mathvariant="normal">∣</mi><mtext> </mtext><mover accent="true"><mi>d</mi><mo>⃗</mo></mover><mtext> </mtext><mi mathvariant="normal">∣</mi><mo><</mo><msub><mi>r</mi><mi>i</mi></msub><mo>+</mo><msub><mi>r</mi><mi>j</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}
\vec{c}_i = \begin{pmatrix}x_i\\ y_i\end{pmatrix} \hspace{16pt} \vec{c}_j = \begin{pmatrix}x_j\\ y_j\end{pmatrix} \\[14pt]
\vec{d} = \vec{c}_{\hspace{.5pt}j} - \vec{c}_{\hspace{.5pt}i}\\[8pt]
\text{collision condition:\hspace{12pt}} |\,\vec{d}\:| < r_i + r_j\\[16pt]
\end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.18488em;vertical-align:-4.34244em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.84244em;"><span style="top:-6.84244em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:0.07142857142857144em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:0.07142857142857144em;"></span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.3675600000000006em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord text"><span class="mord">collision condition:</span><span class="mspace" style="margin-right:1.2000000000000002em;"></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.34244em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (in case of collision ...)</mtext></mstyle></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mstyle mathsize="0.9em"><mtext>By neglecting friction, the balls essentially only collide</mtext></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mstyle mathsize="0.9em"><mtext>along the axis of their centers (direction s), like in </mtext></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mstyle mathsize="0.9em"><mtext>a head-on collision. The velocities perpendicular to </mtext></mstyle></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mstyle mathsize="0.9em"><mtext>the axis of centers (direction t) remain as they were before.</mtext></mstyle></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}
\text{\small\color{Gray} (in case of collision ...)}\\[8pt]
\text{\small By neglecting friction, the balls essentially only collide}\\
\text{\small along the axis of their centers (direction s), like in }\\
\text{\small a head-on collision. The velocities perpendicular to }\\
\text{\small the axis of centers (direction t) remain as they were before.}
\end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.3em;vertical-align:-3.9000000000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4em;"><span style="top:-6.46em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (in case of collision ...)</span></span></span></span><span style="top:-4.159999999999999em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">By neglecting friction, the balls essentially only collide</span></span></span></span><span style="top:-2.6599999999999997em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">along the axis of their centers (direction s), like in </span></span></span></span><span style="top:-1.1599999999999997em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">a head-on collision. The velocities perpendicular to </span></span></span></span><span style="top:0.3400000000000003em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">the axis of centers (direction t) remain as they were before.</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9000000000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<div align="center">contribution: <a href="https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Book%3A_Classical_Mechanics_(Tatum)/05%3A_Collisions/5.04%3A_Oblique_Collisions">libretexts.org</a></div>
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\vec{s} = \frac{1}{|\,\vec{d}\:|} \cdot \vec{d}
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mtext> </mtext><mo>∘</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mtext> </mtext><mo>∘</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>j</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub><mo>⋅</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub><mo>⋅</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>i</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>i</mi></msub><mo>−</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>j</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>j</mi></msub><mo>−</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v_{i,s} &= \vec{s} \:\circ \vec{v}_i & v_{j,s} &= \vec{s} \:\circ \vec{v}_j\\
\vec{v}_{i,s} &= v_{i,s} \cdot \vec{s} & \vec{v}_{j,s} &= v_{j,s} \cdot \vec{s}\\
\vec{v}_{i,t} &= \vec{v}_i - \vec{v}_{i,s} & \vec{v}_{j,t} &= \vec{v}_j - \vec{v}_{j,s}\\
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.500000000000002em;vertical-align:-2.000000000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>m</mi><mi>i</mi></msub><mo>−</mo><msub><mi>m</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msub><mi>m</mi><mi>j</mi></msub><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mrow><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>+</mo><msub><mi>m</mi><mi>j</mi></msub></mrow></mfrac></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>m</mi><mi>j</mi></msub><mo>−</mo><msub><mi>m</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msub><mi>m</mi><mi>i</mi></msub><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mrow><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>+</mo><msub><mi>m</mi><mi>j</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v'_{i,s} &= \frac{v_{i,s}(m_i - m_j) + 2m_jv_{j,s}}{m_i+m_j} & v'_{j,s} &= \frac{v_{j,s}(m_j - m_i) + 2m_iv_{i,s}}{m_i+m_j}\\
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.699108em;vertical-align:-1.0995540000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5995539999999997em;"><span style="top:-3.5995540000000004em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0995540000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5995539999999997em;"><span style="top:-3.5995540000000004em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0995540000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5995539999999997em;"><span style="top:-3.5995540000000004em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0995540000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5995539999999997em;"><span style="top:-3.5995540000000004em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0995540000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>⋅</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>⋅</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi><mo separator="true">,</mo><mi>t</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>i</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi><mo separator="true">,</mo><mi>t</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mrow><mi>j</mi><mo separator="true">,</mo><mi>t</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi><mo separator="true">,</mo><mi>t</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>+</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi><mo separator="true">,</mo><mi>t</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\vec{v}\,'_{\!i,s} &= v'_{i,s} \cdot \vec{s} & \vec{v}\,'_{\!j,s} &= v'_{j,s} \cdot \vec{s}\\
\vec{v}\,'_{\!i,t} &= \vec{v}_{i,t} & \vec{v}\,'_{\!j,t} &= \vec{v}_{j,t}\\
\vec{v}\,'_{\!i} &= \vec{v}\,'_{\!i,s} + \vec{v}\,'_{\!i,t} & \vec{v}\,'_{\!j} &= \vec{v}\,'_{\!j,s} + \vec{v}\,'_{\!j,t}\\
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.569324em;vertical-align:-2.034662em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.534662em;"><span style="top:-4.694662em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mord"><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.19516666666666668em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mord"><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.19516666666666668em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">t</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.034662em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (new velocity)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (new velocity)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (new velocity)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right" columnspacing><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>v</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>i</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>v</mi><mrow><mi>y</mi><mo separator="true">,</mo><mi>i</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>i</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mspace width="1.6em"></mspace><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>v</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>j</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>v</mi><mrow><mi>y</mi><mo separator="true">,</mo><mi>j</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><msubsup><mtext> </mtext><mrow><mtext> ⁣</mtext><mi>j</mi></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\begin{pmatrix}v'_{x,i}\\ v'_{y,i}\end{pmatrix} = \vec{v}\,'_{\!i} \hspace{16pt} \begin{pmatrix}v'_{x,j}\\ v'_{y,j}\end{pmatrix} = \vec{v}\,'_{\!j}\\[20pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.144772em;vertical-align:-1.8223860000000003em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.3223859999999994em;"><span style="top:-4.322386em;"><span class="pstrut" style="height:3.484772em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.484772em;"><span style="top:-3.644772em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.394772em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.394772em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.984772em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mord"><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.19516666666666668em;"></span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:1.6em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.484772em;"><span style="top:-3.644772em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.394772em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.394772em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.984772em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mord"><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mspace mtight" style="margin-right:-0.19516666666666668em;"></span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.8223860000000003em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— ball-wall collisions ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- ball-wall collisions ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">——— ball-wall collisions ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (new velocity ...)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (new velocity ...)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (new velocity ...)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (left and right wall – for each ball)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (left and right wall -- for each ball)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (left and right wall – for each ball)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right" columnspacing><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>v</mi><mi>x</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if</mtext><mspace width="1em"></mspace><mi>x</mi><mo><</mo><mi>r</mi><mtext>  </mtext><mtext>  </mtext><mo>∨</mo><mtext>  </mtext><mtext>  </mtext><mi>w</mi><mo>−</mo><mi>r</mi><mo><</mo><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v''_x =
\begin{cases}
- v'_x, & \text{if}\quad x < r \;\;\vee\;\; w-r < x\\
v'_x, & \text{otherwise}
\end{cases}\\[24pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8100000000000005em;vertical-align:-2.1550000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6550000000000002em;"><span style="top:-4.655em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal">x</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1550000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (top and bottom wall – for each ball)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (top and bottom wall -- for each ball)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (top and bottom wall – for each ball)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right" columnspacing><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>v</mi><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi>v</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if</mtext><mspace width="1em"></mspace><mi>y</mi><mo><</mo><mi>r</mi><mtext>  </mtext><mtext>  </mtext><mo>∨</mo><mtext>  </mtext><mtext>  </mtext><mi>h</mi><mo>−</mo><mi>r</mi><mo><</mo><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>v</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v''_y =
\begin{cases}
- v'_y, & \text{if}\quad y < r \;\;\vee\;\; h-r < y\\
v'_y, & \text{otherwise}
\end{cases}
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.30003em;vertical-align:-1.400015em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.900015em;"><span style="top:-3.9000150000000002em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400015em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<br>
<h2 class="mume-header" id="code">code</h2>
<pre data-role="codeBlock" data-info="js" class="language-javascript"><span class="token keyword">const</span> balls <span class="token operator">=</span> <span class="token punctuation">[</span>
<span class="token punctuation">{</span>
x<span class="token operator">:</span> <span class="token number">30</span><span class="token punctuation">,</span> <span class="token comment">// at the left</span>
y<span class="token operator">:</span> <span class="token number">0.5</span> <span class="token operator">*</span> canvas<span class="token punctuation">.</span><span class="token property-access">h</span><span class="token punctuation">,</span> <span class="token comment">// vertically centered</span>
v_x<span class="token operator">:</span> <span class="token number">10</span><span class="token punctuation">,</span> <span class="token comment">// moving straight to the right</span>
v_y<span class="token operator">:</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token comment">//</span>
m<span class="token operator">:</span> <span class="token number">1</span><span class="token punctuation">,</span>
r<span class="token operator">:</span> <span class="token number">15</span><span class="token punctuation">,</span>
color<span class="token operator">:</span> <span class="token string">'#E91E63'</span><span class="token punctuation">,</span>
<span class="token punctuation">}</span><span class="token punctuation">,</span>
<span class="token punctuation">{</span>
x<span class="token operator">:</span> canvas<span class="token punctuation">.</span><span class="token property-access">w</span> <span class="token operator">-</span> <span class="token number">50</span><span class="token punctuation">,</span> <span class="token comment">// at the right</span>
y<span class="token operator">:</span> <span class="token number">0.5</span> <span class="token operator">*</span> canvas<span class="token punctuation">.</span><span class="token property-access">h</span><span class="token punctuation">,</span> <span class="token comment">// vertically centered</span>
v_x<span class="token operator">:</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token comment">// initially stationary</span>
v_y<span class="token operator">:</span> <span class="token number">0</span><span class="token punctuation">,</span> <span class="token comment">//</span>
m<span class="token operator">:</span> <span class="token number">1</span><span class="token punctuation">,</span>
r<span class="token operator">:</span> <span class="token number">15</span><span class="token punctuation">,</span>
color<span class="token operator">:</span> <span class="token string">'#00BCD4'</span><span class="token punctuation">,</span>
<span class="token punctuation">}</span><span class="token punctuation">,</span>
<span class="token comment">// ...</span>
<span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">function</span> <span class="token function">simulateOneStep</span><span class="token punctuation">(</span><span class="token parameter">dt</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token keyword control-flow">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> ball <span class="token keyword">of</span> balls<span class="token punctuation">)</span> <span class="token punctuation">{</span>
ball<span class="token punctuation">.</span><span class="token property-access">x</span> <span class="token operator">+=</span> dt <span class="token operator">*</span> ball<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">;</span>
ball<span class="token punctuation">.</span><span class="token property-access">y</span> <span class="token operator">+=</span> dt <span class="token operator">*</span> ball<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token function">forEachPair</span><span class="token punctuation">(</span>balls<span class="token punctuation">,</span> <span class="token punctuation">(</span><span class="token parameter">i<span class="token punctuation">,</span> j</span><span class="token punctuation">)</span> <span class="token arrow operator">=></span> <span class="token punctuation">{</span>
<span class="token keyword">const</span> c_i <span class="token operator">=</span> <span class="token punctuation">[</span>i<span class="token punctuation">.</span><span class="token property-access">x</span><span class="token punctuation">,</span> i<span class="token punctuation">.</span><span class="token property-access">y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> c_j <span class="token operator">=</span> <span class="token punctuation">[</span>j<span class="token punctuation">.</span><span class="token property-access">x</span><span class="token punctuation">,</span> j<span class="token punctuation">.</span><span class="token property-access">y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> d <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">subtract</span><span class="token punctuation">(</span>c_j<span class="token punctuation">,</span> c_i<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span><span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">norm</span><span class="token punctuation">(</span>d<span class="token punctuation">)</span> <span class="token operator"><</span> i<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">+</span> j<span class="token punctuation">.</span><span class="token property-access">r</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token keyword">const</span> v_i <span class="token operator">=</span> <span class="token punctuation">[</span>i<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> i<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_j <span class="token operator">=</span> <span class="token punctuation">[</span>j<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> j<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> s <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">normalize</span><span class="token punctuation">(</span>d<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_is_value <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">dotProduct</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_i<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_js_value <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">dotProduct</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_j<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_is <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">scale</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_is_value<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_js <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">scale</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_js_value<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_it <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">subtract</span><span class="token punctuation">(</span>v_i<span class="token punctuation">,</span> v_is<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_jt <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">subtract</span><span class="token punctuation">(</span>v_j<span class="token punctuation">,</span> v_js<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_is_value <span class="token operator">=</span> <span class="token punctuation">(</span>
v_is_value <span class="token operator">*</span> <span class="token punctuation">(</span>i<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">-</span>j<span class="token punctuation">.</span><span class="token property-access">m</span><span class="token punctuation">)</span> <span class="token operator">+</span> <span class="token number">2</span> <span class="token operator">*</span> j<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">*</span> v_js_value
<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span>
i<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">+</span> j<span class="token punctuation">.</span><span class="token property-access">m</span>
<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_js_value <span class="token operator">=</span> <span class="token punctuation">(</span>
v_js_value <span class="token operator">*</span> <span class="token punctuation">(</span>j<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">-</span>i<span class="token punctuation">.</span><span class="token property-access">m</span><span class="token punctuation">)</span> <span class="token operator">+</span> <span class="token number">2</span> <span class="token operator">*</span> i<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">*</span> v_is_value
<span class="token punctuation">)</span> <span class="token operator">/</span> <span class="token punctuation">(</span>
i<span class="token punctuation">.</span><span class="token property-access">m</span> <span class="token operator">+</span> j<span class="token punctuation">.</span><span class="token property-access">m</span>
<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_is <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">scale</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> vˈ_is_value<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_js <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">scale</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> vˈ_js_value<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_it <span class="token operator">=</span> v_it<span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_jt <span class="token operator">=</span> v_jt<span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_i <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">add</span><span class="token punctuation">(</span>vˈ_is<span class="token punctuation">,</span> vˈ_it<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> vˈ_j <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">add</span><span class="token punctuation">(</span>vˈ_js<span class="token punctuation">,</span> vˈ_jt<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">[</span>i<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> i<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span> <span class="token operator">=</span> vˈ_i<span class="token punctuation">;</span>
<span class="token punctuation">[</span>j<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> j<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span> <span class="token operator">=</span> vˈ_j<span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword control-flow">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> ball <span class="token keyword">of</span> balls<span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span>ball<span class="token punctuation">.</span><span class="token property-access">x</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">||</span> canvas<span class="token punctuation">.</span><span class="token property-access">w</span> <span class="token operator">-</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">x</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
ball<span class="token punctuation">.</span><span class="token property-access">v_x</span> <span class="token operator">*=</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span>ball<span class="token punctuation">.</span><span class="token property-access">y</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">||</span> canvas<span class="token punctuation">.</span><span class="token property-access">h</span> <span class="token operator">-</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">y</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
ball<span class="token punctuation">.</span><span class="token property-access">v_y</span> <span class="token operator">*=</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token keyword">function</span> <span class="token function">forEachPair</span><span class="token punctuation">(</span><span class="token parameter">array<span class="token punctuation">,</span> callback</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token keyword control-flow">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator"><</span> array<span class="token punctuation">.</span><span class="token property-access">length</span><span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token comment">// loop every item</span>
<span class="token keyword control-flow">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> j <span class="token operator">=</span> i <span class="token operator">+</span> <span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator"><</span> array<span class="token punctuation">.</span><span class="token property-access">length</span><span class="token punctuation">;</span> j<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span> <span class="token comment">// loop every item after the current item</span>
<span class="token function">callback</span><span class="token punctuation">(</span>array<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">,</span> array<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">,</span> i<span class="token punctuation">,</span> j<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><br>
<h2 class="mume-header" id="discussion-of-the-time-step-equations">discussion of the time step equations</h2>
<ul>
<li>In this level we have utilized the most simple time step equations, like initially in level 2 and level 3.</li>
<li>In contrast to level 2 and level 3, averaging velocities really makes no sense here, because velocities are not gradually changing due to an acceleration. We only instantaneously update velocities (deterministic collisions) on collisions. Between collisions a ball's velocity remains constant - no need to average it xD.</li>
</ul>
<br>
<h2 class="mume-header" id="collision-problems">collision problems</h2>
<ul>
<li>Glitcheees ❤️<div align="center"><img src="img/level-4-billiard-glitch.gif" alt="level-4-billiard-glitch" width="500"></div>
</li>
</ul>
<br>
<ul>
<li>
<p>Problem (1):</p>
<ul>
<li>Balls stick in each other and spin fast xD.</li>
</ul>
</li>
<li>
<p>Reason (1):</p>
<ul>
<li>In our simulation, we strictly process collision sequentially (one after another), even if a ball actually collides (overlaps) with multiple other balls and not just one. The resulting behavior could be named "The Last Collision Wins".</li>
<li>When multiple balls collide in a dense bulk, then the "The Last Collision Wins" rule can push two balls (that would normally bounce off each other) deep into each other (if the collisions with the other balls happen after the collision between the two balls).</li>
<li>If then the outer balls retract, the two inner balls keep sticking in each other, because they repeatedly reflect away from each other and then towards each other again. This is because the collision condition only checks whether the two balls geometrically overlap, which keeps being true because they were pushed so deep into each other that the step away from each other does not suffice to separate them.</li>
<li>And the reason for the balls orbiting around each other is that in an oblique collision the velocity perpendicular to the line of centers is preserved, while the velocity along the line of centers keeps switching direction (effectively cancels out).</li>
</ul>
</li>
<li>
<p>Solution (1):</p>
<ul>
<li>We can simply extend our ball-ball collision condition, to not only check whether the balls are overlapping, but also whether the balls are actually moving towards each other xD.<br>
<br></li>
</ul>
</li>
<li>
<p>Problem (2):</p>
<ul>
<li>Balls sometimes stick in the wall xD.</li>
</ul>
</li>
<li>
<p>Reason (2):</p>
<ul>
<li>When a ball overlaps with the wall, and due to collision with another ball (at the same time) has a velocity directed away from the wall, then our wall collision actually sucks in our ball instead of reflecting it (because we simply reverse the velocity).</li>
<li>And once a ball is two steps in the wall, it can't free itself anymore. It keeps jumping one step forwards and one step backwards, not "leaving" the wall.</li>
</ul>
</li>
<li>
<p>Solution (2):</p>
<ul>
<li>Yo, let's simply extend our ball-wall collision condition, to not only check whether the ball overlaps with the wall, but also whether the ball is actually moving into the wall ^^.</li>
</ul>
</li>
</ul>
<br>
<h2 class="mume-header" id="equations-smallwith-glitch-prevention-only-what-changedsmall">equations <small>(with glitch prevention) (only what changed)</small></h2>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— ball-ball collisions ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- ball-ball collisions ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">——— ball-ball collisions ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (for every pair of balls, i and j, ...)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (for every pair of balls, i and j, ...)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (for every pair of balls, i and j, ...)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mi>i</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>i</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>i</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mspace width="1.6em"></mspace><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mi>j</mi></msub><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>j</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>j</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>d</mi><mo>⃗</mo></mover><mo>=</mo><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mrow><mspace width="0.05em"></mspace><mi>j</mi></mrow></msub><mo>−</mo><msub><mover accent="true"><mi>c</mi><mo>⃗</mo></mover><mrow><mspace width="0.05em"></mspace><mi>i</mi></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">∣</mi><mtext> </mtext><mover accent="true"><mi>d</mi><mo>⃗</mo></mover><mtext> </mtext><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>⋅</mo><mover accent="true"><mi>d</mi><mo>⃗</mo></mover></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{gathered}
\vec{c}_i = \begin{pmatrix}x_i\\ y_i\end{pmatrix} \hspace{16pt} \vec{c}_j = \begin{pmatrix}x_j\\ y_j\end{pmatrix} \\[14pt]
\vec{d} = \vec{c}_{\hspace{.5pt}j} - \vec{c}_{\hspace{.5pt}i}\\[8pt]
\vec{s} = \frac{1}{|\,\vec{d}\:|} \cdot \vec{d}
\end{gathered}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.686320000000002em;vertical-align:-4.093160000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.59316em;"><span style="top:-6.59316em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:1.6em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.1325600000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left right left" columnspacing="0em 1em 0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>i</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mrow><mi>y</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mspace width="1.6em"></mspace><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>j</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>v</mi><mrow><mi>y</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mtext> </mtext><mo>∘</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mover accent="true"><mi>s</mi><mo>⃗</mo></mover><mtext> </mtext><mo>∘</mo><msub><mover accent="true"><mi>v</mi><mo>⃗</mo></mover><mi>j</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
\vec{v}_i &= \begin{pmatrix}v_{x,i}\\ v_{y,i}\end{pmatrix} &\hspace{16pt} \vec{v}_j &= \begin{pmatrix}v_{x,j}\\ v_{y,j}\end{pmatrix} \\[14pt]
v_{i,s} &= \vec{s} \:\circ \vec{v}_i & v_{j,s} &= \vec{s} \:\circ \vec{v}_j\\[14pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.410000000000002em;vertical-align:-2.955000000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.455000000000001em;"><span style="top:-5.455000000000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
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-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
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style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.955000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.455000000000001em;"><span style="top:-5.455000000000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span><span style="top:-2.5549999999999997em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.955000000000001em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.455000000000001em;"><span style="top:-5.455000000000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mspace" style="margin-right:1.6em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.5549999999999997em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.955000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.455000000000001em;"><span style="top:-5.455000000000001em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span><span style="top:-2.5549999999999997em;"><span class="pstrut" style="height:3.45em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.17994em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∘</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.20772em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.955000000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtable rowspacing="0.24999999999999992em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mtext>collision condition:</mtext><mspace width="1.2000000000000002em"></mspace></mrow><mi mathvariant="normal">∣</mi><mtext> </mtext><mover accent="true"><mi>d</mi><mo>⃗</mo></mover><mtext> </mtext><mi mathvariant="normal">∣</mi><mo><</mo><msub><mi>r</mi><mi>i</mi></msub><mo>+</mo><msub><mi>r</mi><mi>j</mi></msub><mspace width="1em"></mspace><mo>∧</mo><mspace width="1em"></mspace><msub><mi>v</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub><mo>></mo><msub><mi>v</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>s</mi></mrow></msub></mrow></mstyle></mtd></mtr></mtable><mspace linebreak="newline" height="-1em"></mspace><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mspace linebreak="newline" height="2.4000000000000004em"></mspace></mrow><annotation encoding="application/x-tex">\begin{gathered}
\text{collision condition:\hspace{12pt}} |\,\vec{d}\:| < r_i + r_j \quad\wedge\quad v_{i,s} > v_{j,s}\\[20pt]
\end{gathered}\\[-10pt]
...\\[24pt]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.63744em;vertical-align:-1.5687200000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.06872em;"><span style="top:-4.09128em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">collision condition:</span><span class="mspace" style="margin-right:1.2000000000000002em;"></span></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9774399999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span><span style="top:-3.26344em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.06882999999999997em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5
3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11
10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63
-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1
-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59
H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359
c-16-25.333-24-45-24-59z"/></svg></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">s</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5687200000000001em;"><span></span></span></span></span></span></span></span></span><span class="mspace newline" style="margin-top:-1em;"></span><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span><span class="mspace newline" style="margin-top:2.4000000000000004em;"></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>——— ball-wall collisions ———</mtext></mrow><annotation encoding="application/x-tex">\text{--------- ball-wall collisions ---------}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">——— ball-wall collisions ———</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (left and right wall – for each ball)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (left and right wall -- for each ball)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (left and right wall – for each ball)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right" columnspacing><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>v</mi><mi>x</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if</mtext><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo><</mo><mi>r</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mi>x</mi></msub><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mtext>  </mtext><mtext>  </mtext><mo>∨</mo><mtext>  </mtext><mtext>  </mtext><mo stretchy="false">(</mo><mi>w</mi><mo>−</mo><mi>r</mi><mo><</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mn>0</mn><mo><</mo><msub><mi>v</mi><mi>x</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>v</mi><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v''_x =
\begin{cases}
- v'_x, & \text{if}\quad (x < r) \wedge (v_x < 0) \;\;\vee\;\; (w-r < x) \wedge (0 < v_x)\\
v'_x, & \text{otherwise}
\end{cases}\\[24pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8100000000000005em;vertical-align:-2.1550000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6550000000000002em;"><span style="top:-4.655em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1550000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mstyle mathcolor="Gray"><mtext> (top and bottom wall – for each ball)</mtext></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small\color{Gray} (top and bottom wall -- for each ball)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5" style="color:Gray;"> (top and bottom wall – for each ball)</span></span></span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right" columnspacing><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>v</mi><mi>y</mi><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msubsup><mi>v</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mtext>if</mtext><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>y</mi><mo><</mo><mi>r</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>v</mi><mi>y</mi></msub><mo><</mo><mn>0</mn><mo stretchy="false">)</mo><mtext>  </mtext><mtext>  </mtext><mo>∨</mo><mtext>  </mtext><mtext>  </mtext><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mi>r</mi><mo><</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mn>0</mn><mo><</mo><msub><mi>v</mi><mi>y</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>v</mi><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>otherwise</mtext></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}
v''_y =
\begin{cases}
- v'_y, & \text{if}\quad (y < r) \wedge (v_y < 0) \;\;\vee\;\; (h-r < y) \wedge (0 < v_y)\\
v'_y, & \text{otherwise}
\end{cases}\\[24pt]
\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8100000000000005em;vertical-align:-2.1550000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6550000000000002em;"><span style="top:-4.655em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">if</span></span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∨</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord">otherwise</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1550000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<br>
<h2 class="mume-header" id="code-smallwith-glitch-prevention-only-what-changedsmall">code <small>(with glitch prevention) (only what changed)</small></h2>
<pre data-role="codeBlock" data-info="js" class="language-javascript"><span class="token comment">// ...</span>
<span class="token keyword">function</span> <span class="token function">simulateOneStep</span><span class="token punctuation">(</span><span class="token parameter">dt</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token comment">// ...</span>
<span class="token function">forEachPair</span><span class="token punctuation">(</span>balls<span class="token punctuation">,</span> <span class="token punctuation">(</span><span class="token parameter">i<span class="token punctuation">,</span> j</span><span class="token punctuation">)</span> <span class="token arrow operator">=></span> <span class="token punctuation">{</span>
<span class="token keyword">const</span> c_i <span class="token operator">=</span> <span class="token punctuation">[</span>i<span class="token punctuation">.</span><span class="token property-access">x</span><span class="token punctuation">,</span> i<span class="token punctuation">.</span><span class="token property-access">y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> c_j <span class="token operator">=</span> <span class="token punctuation">[</span>j<span class="token punctuation">.</span><span class="token property-access">x</span><span class="token punctuation">,</span> j<span class="token punctuation">.</span><span class="token property-access">y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> d <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">subtract</span><span class="token punctuation">(</span>c_j<span class="token punctuation">,</span> c_i<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> s <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">normalize</span><span class="token punctuation">(</span>d<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_i <span class="token operator">=</span> <span class="token punctuation">[</span>i<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> i<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_j <span class="token operator">=</span> <span class="token punctuation">[</span>j<span class="token punctuation">.</span><span class="token property-access">v_x</span><span class="token punctuation">,</span> j<span class="token punctuation">.</span><span class="token property-access">v_y</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_is_value <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">dotProduct</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_i<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword">const</span> v_js_value <span class="token operator">=</span> <span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">dotProduct</span><span class="token punctuation">(</span>s<span class="token punctuation">,</span> v_j<span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span>
<span class="token punctuation">(</span><span class="token maybe-class-name">Vector</span><span class="token punctuation">.</span><span class="token method function property-access">norm</span><span class="token punctuation">(</span>d<span class="token punctuation">)</span> <span class="token operator"><</span> i<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">+</span> j<span class="token punctuation">.</span><span class="token property-access">r</span><span class="token punctuation">)</span> <span class="token operator">&&</span>
<span class="token punctuation">(</span>v_is_value <span class="token operator">></span> v_js_value<span class="token punctuation">)</span>
<span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token comment">// ...</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
<span class="token keyword control-flow">for</span> <span class="token punctuation">(</span><span class="token keyword">let</span> ball <span class="token keyword">of</span> balls<span class="token punctuation">)</span> <span class="token punctuation">{</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span>
ball<span class="token punctuation">.</span><span class="token property-access">x</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">&&</span> ball<span class="token punctuation">.</span><span class="token property-access">v_x</span> <span class="token operator"><</span> <span class="token number">0</span> <span class="token operator">||</span>
canvas<span class="token punctuation">.</span><span class="token property-access">w</span> <span class="token operator">-</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">x</span> <span class="token operator">&&</span> ball<span class="token punctuation">.</span><span class="token property-access">v_x</span> <span class="token operator">></span> <span class="token number">0</span>
<span class="token punctuation">)</span> <span class="token punctuation">{</span>
ball<span class="token punctuation">.</span><span class="token property-access">v_x</span> <span class="token operator">*=</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token keyword control-flow">if</span> <span class="token punctuation">(</span>
ball<span class="token punctuation">.</span><span class="token property-access">y</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator">&&</span> ball<span class="token punctuation">.</span><span class="token property-access">v_y</span> <span class="token operator"><</span> <span class="token number">0</span> <span class="token operator">||</span>
canvas<span class="token punctuation">.</span><span class="token property-access">h</span> <span class="token operator">-</span> ball<span class="token punctuation">.</span><span class="token property-access">r</span> <span class="token operator"><</span> ball<span class="token punctuation">.</span><span class="token property-access">y</span> <span class="token operator">&&</span> ball<span class="token punctuation">.</span><span class="token property-access">v_y</span> <span class="token operator">></span> <span class="token number">0</span>
<span class="token punctuation">)</span> <span class="token punctuation">{</span>
ball<span class="token punctuation">.</span><span class="token property-access">v_y</span> <span class="token operator">*=</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</pre><br>
<h2 class="mume-header" id="inaccuracy-problem">inaccuracy problem</h2>
<ul>
<li>(1) Collisions of multiple balls at once are not realistic in the current simulation.
<ul>
<li>In this example the balls should actually move on symmetric paths. But they don't.</li>
</ul>
<div align="center"><img src="img/level-4-unsymmetric-billiard-kick-off.gif" alt="level-4-unsymmetric-billiard-kick-off" width="500"></div>
</li>
<li>Reason:
<ul>
<li>In our simulation, we process collision sequentially (one after another). This "The Last Collision Wins" rule explains why the balls aren't moving symmetrically right from the start.</li>
</ul>
</li>
<li>Weakening Factor:
<ul>
<li>Normally, collisions are so fast, that we can't really see if they should have lead to different results, because the balls actually collided simultaneously instead of one after another.</li>
<li>Normally, everything looks fine to the observer of the simulation.</li>
<li>It's only in special cases, like the kickoff at the start, that the inaccuracy becomes obvious. Only under these special conditions we have clear predictions of the collision results.</li>
</ul>
</li>
<li>Solution:
<ul>
<li>We will tackle that in the next two upcoming levels 😛</li>
</ul>
</li>
</ul>
<br>
<h2 class="mume-header" id="working-example-smallfor-all-variantssmall">working example <small>(for all variants)</small></h2>
<table>
<thead>
<tr>
<th></th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="https://github.com/pitizzzle/physics-simulations-balls/blob/main/code/level-4-ball-deterministic-collisions.html">Code</a></td>
<td><a href="https://pitizzzle.github.io/simulate-ball-physics/code/level-4-ball-deterministic-collisions.html">Code Live</a></td>
</tr>
</tbody>
</table>
</div>
</body></html>