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384 lines
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HTML
384 lines
13 KiB
HTML
<!DOCTYPE html>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Momentum</h1>
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<p>Momentum is how much something wants to <b>keep moving</b> in the same direction.</p>
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<p class="center large">This truck would be hard to stop ...<br>
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<img src="images/truck.jpg" alt="truck" height="211" width="360"><br>
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... it has a <b>lot of momentum</b>.</p>
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<p class="center larger"><b>Faster?</b> More momentum! <b><br>
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Heavier?</b> More momentum!</p>
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<p>Momentum is mass times velocity.<br>
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The symbol is <b>p</b>:</p>
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<p class="center larger"><b>p</b> = m <b>v</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/car-moving.jpg" alt="car moving" height="89" width="250"></p>
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<h3>Example: What is the momentum of a 1500 kg car going at highway speed of <b>28 m/s</b> (about 100 km/h or 60 mph)?</h3>
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<br> <p class="so"><b>p</b> = m <b>v</b></p>
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<p class="so"><b>p</b> = 1500 kg × <b>28 m/s</b></p>
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<p class="so"><b>p</b> = 42,000 kg m/s</p>
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</div>
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<p>The <a href="../measure/metric-system.html">unit</a> for momentum is:</p>
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<ul>
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<li>kg m/s (kilogram meter per second), or</li>
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<li>N s (Newton second)</li>
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</ul>
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<p>They are the same! <b>1 kg m/s = 1 N s</b></p>
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<p>We will use both here.</p>
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<p>More examples:</p>
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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>
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<td> </td>
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<th>Mass</th>
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<th>Speed</th>
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<th>Momentum</th>
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</tr>
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<tr>
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<th>Bullet (9 mm)</th>
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<td style="text-align:center;">7.5 g<br>
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0.0075 kg</td>
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<td style="text-align:center;"><br>
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1000 m/s</td>
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<td style="text-align:center;"><br>
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0.0075 × 1000 = <b>7.5 kg m/s</b></td>
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</tr>
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<tr>
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<th>Tennis Ball</th>
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<td style="text-align:center;">57 g<br>
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0.057 kg</td>
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<td style="text-align:center;"><br>
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50 m/s</td>
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<td style="text-align:center;"><br>
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0.057 × 50 = <b>2.85 kg m/s</b></td>
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</tr>
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<tr>
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<th>Soccer Ball</th>
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<td style="text-align:center;">16 oz<br>
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0.45 kg</td>
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<td style="text-align:center;">100 km/h<br>
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28 m/s</td>
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<td style="text-align:center;"><br>
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0.45 × 28 = <b>12.6 kg m/s</b></td>
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</tr>
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<tr>
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<th>Basket Ball</th>
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<td style="text-align:center;">22 oz<br>
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0.6 kg</td>
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<td style="text-align:center;"><br>
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3 m/s</td>
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<td style="text-align:center;"><br>
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0.6 × 3 = <b>1.8 kg m/s</b></td>
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</tr>
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<tr>
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<th>Hammer</th>
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<td style="text-align:center;">400 g<br>
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0.4 kg
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</td>
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<td style="text-align:center;"><br>
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7 m/s</td>
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<td style="text-align:center;"><br>
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0.4 × 7 = <b>2.8 kg m/s</b></td>
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</tr>
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<tr>
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<th>Runner</th>
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<td style="text-align:center;"><br>
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80 kg</td>
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<td style="text-align:center;">9 km/h<br>
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2.5 m/s</td>
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<td style="text-align:center;"><br>
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80 × 2.5 = <b>200 kg m/s</b></td>
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</tr>
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<tr>
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<th>Car</th>
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<td style="text-align:center;"><br>
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1500 kg</td>
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<td style="text-align:center;">100 km/h<br>
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28 m/s</td>
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<td style="text-align:center;"><br>
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1500 × 28 = <b>42,000 kg m/s</b></td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="def">
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<p>Momentum has <b>direction</b>: the <b>exact same direction</b> as the <a href="../measure/speed-velocity.html">velocity</a>.</p>
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<p>But many examples here only use speed (velocity without direction) to keep it simple.</p>
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</div>
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<br>
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<p style="float:left; margin: 0 10px 5px 0;"><a href="momentum-animation.html"><img src="images/momentum-animation.gif" alt="momentum animation thumbnail" height="112" width="200"></a></p>
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<h3>Animation</h3>
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<p>Play with <a href="momentum-animation.html">momentum in this animation</a>.</p>
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<h2>Impulse</h2>
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<p>Impulse is change in momentum. <span class="center">Δ is the symbol for "change in"</span>, so:</p>
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<p class="center larger">Impulse is Δp</p>
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<p>Force can be calculated from the change in momentum over time (called the "time rate of change" of momentum):</p>
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<p class="center larger">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/brick-wall.jpg" alt="brick wall" height="109" width="200"><br>
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<img src="images/padding.jpg" alt="padding" height="110" width="200"></p>
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<h3>Example: You are 60 kg and run at 3 m/s into a wall.<br>
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<br>
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The wall stops you in 0.05 s. What is the force?<br>
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<br>
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The wall is then padded and stops you in 0.2 s. What is the force?</h3>
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<p>First calculate the impulse:</p>
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<p class="so">Δp = m v</p>
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<p class="so">Δp = 60 kg x 3 m/s</p>
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<p class="so">Δp = 180 kg m/s</p>
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<p>Stopping in 0.05 s:</p>
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<p class="so">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></p>
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<p class="so">F = <span class="intbl">
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<em>180 kg m/s</em>
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<strong>0.05 s</strong>
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</span> = 3600 N</p>
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<p>Stopping in 0.2 s:</p>
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<p class="so">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></p>
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<p class="so">F = <span class="intbl">
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<em>180 kg m/s</em>
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<strong>0.2 s</strong>
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</span> = 900 N</p>
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<p>Stopping at a slower rate has much less force!</p>
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<ul>
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<li>And that is why padding works so well</li>
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<li>And also why crash helmets save lives</li>
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<li>And why cars have crumple zones</li>
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</ul>
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</div>
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<div class="def">
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<h3>Q: Isn't force normally calculated using <span class="large">F = ma</span> ?<br>
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A: Well, <span class="center large">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></span> is the <b>same thing</b>, just a different form:</h3>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:right;">Start with:</td>
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<td style="width:20px;"> </td>
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<td><span class="larger">F = ma</span></td>
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</tr>
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<tr>
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<td style="text-align:right;">Acceleration is change in velocity <b>v</b> over time <b>t</b>:</td>
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<td> </td>
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<td><span class="larger">F = m<span class="intbl">
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<em>Δv</em>
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<strong>Δt</strong>
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</span></span></td>
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</tr>
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<tr>
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<td style="text-align:right;">Rearrange to:</td>
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<td> </td>
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<td><span class="larger">F = <span class="intbl">
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<em>Δmv</em>
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<strong>Δt</strong>
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</span></span></td>
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</tr>
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<tr>
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<td style="text-align:right;">And <b>Δmv</b> is change in momentum:</td>
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<td> </td>
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<td><span class="larger">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></span></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Impulse From Force</h2>
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<p>We can rearrange:</p>
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<p class="center larger">F = <span class="intbl">
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<em>Δp</em>
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<strong>Δt</strong>
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</span></p>
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<p>Into:</p>
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<p class="center larger">Δp = F Δt</p>
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<p>So we can calculate the Impulse (the change in momentum) from force applied for a period of time.</p>
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<div class="example">
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<h3>Example: A ball is hit with a 300 N force. High speed cameras show the contact lasted for 0.02 s. What was the impulse?</h3>
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<p class="center larger">Δp = F Δt</p>
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<p class="center larger">Δp = 300 N × 0.02 s</p>
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<p class="center larger">Δp = 6 N s</p>
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</div>
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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="system" height="124" width="169"></p>
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<p class="words"><b>Closed System</b>: where 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="energy-work.html">Energy</a> is conserved (it also can change form, to light, to heat and so on)</li>
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<li>And <b>Momentum</b> is also 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 is a Vector</h2>
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<p>Momentum is a <a href="../algebra/vectors.html">vector</a>: it has size AND direction.</p>
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<p class="center"><img src="../algebra/images/vector-mag-dir.svg" alt="vector magnitude and direction" height="137" width="267"></p>
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<p>Sometimes we don't mention the direction, but other times it is important!</p>
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<h3>One Dimension</h3>
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<p>A question may have only one dimension, and all we need is positive or negative momentum:</p>
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<p class="center"><img src="images/neg-pos.svg" alt="negative positive" height="35" width="280"></p>
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<h3>Two or More Dimensions</h3>
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<p>Questions can be in two (or more) dimensions like this one:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/ball-bounce.gif" alt="ball bounces at 50 degrees" height="97" width="166"></p>
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<h3>Example: A pool ball bounces!<br>
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<br>
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It hits the edge with a velocity of <b>8 m/s at 50°</b>, and bounces off at the same speed and reflected angle.<br>
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<br>
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It weighs 0.16 kg. What is the change in momentum?</h3>
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<p>Let's break the velocity into x and y parts. Before the bounce:</p>
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<ul>
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<li>v<sub>x</sub> = 8 × cos(50°) <i>...going along</i></li>
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<li>v<sub>y</sub> = 8 × sin(50°) <i>...going up</i></li>
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</ul>
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<p>After the bounce:</p>
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<ul>
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<li>v<sub>x</sub> = 8 × cos(50°) <i>...going along</i></li>
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<li>v<sub>y</sub> = 8 × −sin(50°) <i>...going down</i></li>
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</ul>
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<p>The x-velocity does not change, but the y-velocity changes by:</p>
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<p class="center large">Δv<sub>y</sub> = (8+8) × sin(50°)<br>
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= 16 × sin(50°)</p>
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<p>And the change in momentum is:</p>
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<p class="center larger"><b>Δp</b> = m <b>Δv</b></p>
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<p class="center larger">Δp = 0.16 kg × 16 × sin(50°) m/s</p>
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||
<p class="center larger">Δp = 1.961... kg m/s</p>
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</div>
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||
|
||
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||
<p>
|
||
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||
</p>
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<div class="fun">
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<h3>Footnote: The formula</h3>
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||
<p class="center"><span class="large"><b>p</b> = m <b>v</b></span><br>
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||
<i>Momentum is mass times velocity</i></p>
|
||
<p>is not the full story!</p>
|
||
<p>It is a wonderful and useful formula for <b>normal every day use</b>, but when we look at the atomic scale things don't actually collide. They interact from a distance through electro-magnetic fields.</p>
|
||
<p>And the interaction does not need mass, because light (which has no mass) can have momentum.</p>
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||
</div>
|
||
<p> </p>
|
||
<div class="questions">11985, 17629, 11991, 17630, 17636, 11993, 17635, 17640, 17643, 17648</div>
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<div class="related">
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<a href="collisions.html">Collisions</a>
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<a href="../measure/metric-mass.html">Mass</a>
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