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<h1 class="center">Collisions</h1>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/newtons-cradle.jpg" alt="newtons-cradle" height="189" width="300"></p>
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<p>A collision is when two objects strike each other over a short space of time.</p>
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<p>The <a href="momentum.html">momentum</a> of each object can change, but the total momentum does not. We say the momentum is <b>conserved</b> (the total stays the same).</p>
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<h2>Momentum is Conserved</h2>
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<p class="words"><b>Conserved</b>: the total stays the same (within a closed system).</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/closed-system.svg" alt="closed system" height="124" width="169"></p>
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<p class="words"><b>Closed System</b>: nothing transfers in or out, and no external force acts on it.</p>
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<div class="fun">
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<h3>In our Universe:</h3>
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<ul>
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<li><a href="../measure/metric-mass.html">Mass</a> is conserved (it can change form, be moved around, cut up or joined together, but the total mass stays the same over time)</li>
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<li><a href="momentum.html">Momentum</a> is also conserved, and</li>
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<li><a href="energy-work.html">Energy</a> is conserved</li>
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</ul>
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</div>
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<p><i>Note: At an atomic level Mass and Energy can be converted via E=mc<sup>2</sup>, but nothing gets lost.</i></p>
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<h2>Momentum and Kinetic Energy</h2>
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<p>You might like to read the page on <a href="energy-work.html">Energy</a> first.</p>
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<p>Kinetic Energy (KE) is the energy of motion:</p>
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<p class="center large">KE = ½ m v<sup>2</sup></p>
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<p>Where:</p>
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<ul>
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<li>KE is the Kinetic Energy in Joules (J, or kg m<sup>2</sup>/s<sup>2</sup>)</li>
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<li>m is mass (in kg)</li>
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<li>v is velocity (in m/s)</li>
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</ul>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/balls-2.svg" alt="1kg ball at 20m/s vs 10kg ball at 2m/s" height="191" width="236"></p>
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<h3>Example: A 1kg ball travels at 20 m/s, and a 10 kg ball travels at 2 m/s.<br>
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<br>
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What is the momentum and KE of each?</h3>
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<h4>Momentum</h4>
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<p>For the 1 kg ball at 20 m/s:</p>
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<p class="so"><b>p</b> = m <b>v</b></p>
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<p class="so"><b>p</b> = 1 kg × <b>20 m/s</b></p>
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<p class="so"><b>p</b> = 20 kg m/s</p>
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<p>For the 10 kg ball at 2 m/s:</p>
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<p class="so"><b>p</b> = 10 kg × <b>2 m/s</b></p>
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<p class="so"><b>p</b> = 20 kg m/s</p>
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<p>The momentum is the same for each.</p>
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<h4>Kinetic Energy (KE)</h4>
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<p>For the 1 kg ball at 20 m/s:</p>
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<p class="so">KE = ½ m v<sup>2</sup></p>
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<p class="so">KE = ½ × 1 kg × (20 m/s)<sup>2</sup></p>
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<p class="so">KE = 200 kg m<sup>2</sup>/s<sup>2</sup> = 200 J</p>
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<p>For the 10 kg ball at 2 m/s:</p>
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<p class="so">KE = ½ × 10 kg × (2 m/s)<sup>2</sup></p>
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<p class="so">KE = 20 kg m<sup>2</sup>/s<sup>2</sup> = 20 J</p>
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<p>The KE of the small ball is <b>much higher</b>!</p>
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<h4>Summary</h4>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th> </th>
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<th>1 kg at 20 m/s</th>
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<th>10 kg at 2 m/s</th>
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</tr>
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<tr style="text-align:center;">
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<th>Momentum:</th>
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<td>20 kg m/s</td>
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<td>20 kg m/s</td>
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</tr>
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<tr style="text-align:center;">
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<th>KE:</th>
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<td>200 J</td>
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<td>20 J</td>
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</tr>
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</tbody></table>
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</div>
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<p>So the momentum can be the same while the KE is very different.</p>
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<p>Because KE uses <b>velocity squared</b>.</p>
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</div>
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<h2>Inelastic vs Elastic Collisions</h2>
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<ul class="larger">
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<li><b>Inelastic</b> collisions are mushy (like dough balls)</li>
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<li><b>Elastic</b> collisions are bouncy (like rubber balls)</li>
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</ul>
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<p class="center"><img src="images/inelastic-elastic.svg" alt="inelastic dough vs elastic bouncy ball" height="112" width="327"></p>
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<p>In a <i>perfectly</i> Inelastic collision:</p>
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<ul>
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<li>the objects stick together and end up sharing a new velocity</li>
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<li>the objects get deformed by the collision, so</li>
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<li>Kinetic Energy is lost (it gets converted into heat, light and sound)</li>
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</ul>
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<p>In a <i>perfectly</i> Elastic collision the objects:</p>
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<ul>
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<li>bounce perfectly off each other</li>
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<li>the total Kinetic Energy stays exactly the same</li>
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</ul>
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<p class="center"><img src="images/inelastic-elastic-drop.svg" alt="inelastic elastic drop" height="164" width="341"></p>
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<p class="center">Collisions are typically <b>in between inelastic and elastic</b>.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/tennis-ball-bounce.jpg" alt="tennis ball bounces" height="201" width="200"></p>
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<h3>Example: drop a tennis ball</h3>
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<p>It won't bounce back to the same height.</p>
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<p>Because some kinetic energy is lost on the bounce (and a little is lost due to air resistance)</p>
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<p>So the bounce is slightly inelastic (but mostly elastic).</p>
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</div>
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<p>We can have a scale of 0 (Inelastic) to 1 (Elastic). Try the <a href="momentum-animation.html">Momentum Animation</a> to see for yourself.</p>
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<p>And in the <a href="../numbers/gravity-freeplay.html">Gravity Animation</a>, the collisions are either inelastic (two objects smashing into each other) or elastic (the objects swing around each other and head off again).</p>
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<div class="example">
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<h3>Example: A railcar weighing 25,000 kg is rolling at 3 m/s east and hooks onto the back of a locomotive weighing 190,000 kg rolling 1 m/s east. What is the combined new velocity?</h3>
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<p class="center"><img src="images/rail-car-loco.gif" alt="rail car loco" height="54" width="520"></p>
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<p>The collision is <b>inelastic</b>, as the coupling locks the railcar and loco together.</p>
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<p>Momentum of Railcar</p>
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<p class="so"><b>p<sub>car</sub></b> = 25,000 kg × <b>3 m/s</b></p>
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<p class="so"><b>p</b><b><sub>car</sub></b> = 75,000 kg m/s</p>
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<p>Momentum of Loco</p>
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<p class="so"><b>p<sub>loco</sub></b> = 190,000 kg × <b>1 m/s</b></p>
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<p class="so"><b>p<sub>loco</sub></b> = 190,000 kg m/s</p>
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<p><b>Momentum is conserved</b>, so the combined momentum is the same as when separated:</p>
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<p class="so"><b>p<sub>tot</sub></b> = m<sub>tot</sub> <b>v</b></p>
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<p>Which can be rearranged to:</p>
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<p class="so">v = <span class="intbl"><em><b>p<sub>tot</sub></b></em><strong>m<sub>tot</sub></strong></span></p>
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<p>The totals are:</p>
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<p class="so"><b>p<sub>tot</sub></b> = 75,000 kg m/s + 190,000 kg m/s = 265,000 kg m/s</p>
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<p class="so"><b>m<sub>tot</sub></b> = 25,000 kg + 190,000 kg = 215,000 kg</p>
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<p>Solving:</p>
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<p class="so"><b>v</b> = <span class="intbl"><em><b>265,000 kg m/s</b></em>
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<strong>215,000 kg</strong></span></p>
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<p class="so"><b>v</b> = 1.2326... m/s</p>
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</div>
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<h2>General Formulas</h2>
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<p>Inelastic is a relatively easy case, but for other cases we can use these general formulas:</p>
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<p class="center large"><b>new v<sub>a</sub></b> = <span class="intbl">
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<em>elast × m<sub>b</sub>(v<sub>b</sub> − v<sub>a</sub>) + m<sub>a</sub>v<sub>a</sub> + m<sub>b</sub>v<sub>b</sub></em>
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<strong>m<sub>a</sub> + m<sub>b</sub></strong>
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</span></p>
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<p class="center large"><b>new v<sub>b</sub></b> = <span class="intbl">
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<em>elast × m<sub>a</sub>(v<sub>a</sub> − v<sub>b</sub>) + m<sub>a</sub>v<sub>a</sub> + m<sub>b</sub>v<sub>b</sub></em>
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<strong>m<sub>a</sub> + m<sub>b</sub></strong>
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</span></p>
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<p>Where:</p>
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<ul>
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<li>elast = "Coefficient of Restitution", 0=inelastic, 1=elastic (or any value in between)</li>
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<li>v<sub>a</sub> and m<sub>a</sub> are the velocity and mass of object a</li>
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<li>v<sub>b</sub> and m<sub>b</sub> are the velocity and mass of object b</li>
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</ul>
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<div class="example">
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<h3>Example (continued): Instead of getting hooked, they <b>bounce perfectly</b> off each other, what are the new velocities?</h3>
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<p class="center"><img src="images/rail-car-loco.gif" alt="rail car loco" height="54" width="520"></p>
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<p>The collision is perfectly <b>elastic</b>. We can use an "elast" value of 1:</p>
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<p class="center larger"><b>new v<sub>a</sub></b> = <span class="intbl">
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<em>elast × m<sub>b</sub>(v<sub>b</sub> − v<sub>a</sub>) + m<sub>a</sub>v<sub>a</sub> + m<sub>b</sub>v<sub>b</sub></em>
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<strong>m<sub>a</sub> + m<sub>b</sub></strong>
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</span></p>
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<p class="center larger"><b>new v<sub>b</sub></b> = <span class="intbl">
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<em>elast × m<sub>a</sub>(v<sub>a</sub> − v<sub>b</sub>) + m<sub>a</sub>v<sub>a</sub> + m<sub>b</sub>v<sub>b</sub></em>
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<strong>m<sub>a</sub> + m<sub>b</sub></strong>
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</span></p>
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<p>Put in the values we know:</p>
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<p class="center larger"><b>new v<sub>a</sub></b> = <span class="intbl">
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<em>1 × 190 × (1 − 3) + 25×3+ 190×1</em>
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<strong>25 + 190</strong>
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</span></p>
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<p class="center larger"><b>new v<sub>b</sub></b> = <span class="intbl">
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<em>1 × 25 × (3 − 1) + 25×3 + 190×1</em>
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<strong>25 + 190</strong>
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</span></p>
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<p>Calculate:</p>
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<p class="center larger"><b>new v<sub>a</sub></b> = −0.5349...</p>
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<p class="center larger"><b>new v<sub>b</sub></b> = 1.4651...</p>
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<p>The railcar is now moving <b>backwards</b> at about 0.5 m/s, and the loco moving forwards at about 1.5 m/s</p>
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<div class="fun">
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<p><b>Is momentum conserved?</b> The momentum values are <b>now</b>:</p>
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<p class="so"><b>p</b><b><sub>car</sub></b> = 25,000 kg × <b>0.5349... m/s</b> = −13,372 kg m/s</p>
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<p class="so"><b>p<sub>loco</sub></b> = 190,000 kg × <b>1.4651... m/s</b> = 278,372 kg m/s</p>
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<p class="so"><b>p<sub>tot</sub></b> = −13,372 kg m/s + 278,372 kg m/s = 265,000 kg m/s</p>
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<p>That is the same total value as in the earlier example, so yes, momentum is conserved.</p>
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</div>
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</div>
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<p> </p>
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<div class="questions">11995, 11997, 17650, 11998, 12000, 12002, 12004, 17652, 17655, 17658</div>
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<div class="related">
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<a href="momentum.html">Momentum</a>
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<a href="gravity-ball.html">Ball Physics Animation</a>
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<a href="index.html">Physics Index</a>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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