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384 lines
13 KiB
HTML
384 lines
13 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/activity/olympic-athletics-track.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:05 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Activity: The Olympic Athletics Track</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: The Olympic
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Athletics Track</h1>
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<p> </p>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td><img src="images/staggered-start.jpg" alt="staggered start" height="161" width="275"></td>
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<td class="larger">Have you ever watched some of the races in the Olympic Games and wondered why the athletes don't all start from the same part of the track?</td>
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</tr>
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</tbody></table>
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<p class="larger">It is called a "staggered start".</p>
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<h2>Why
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a Staggered Start?</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/olympic-long-short.svg" alt="staggered start shorter vs longer"></p>
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<p>If
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they all started from the same line, then the athletes in the <b>outer</b> lanes would have to run <b>further</b> than the athletes in the inner lanes, because of the <b>semicircles</b> at the top and bottom of the track.</p>
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<p>So each lane has to have a special starting position so they all have to run the same distance.</p>
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<p><strong>Let's learn how to calculate the correct positions
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for the 400 m running race</strong></p>
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<h2>How Far?</h2>
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<h3>How
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far does each athlete run when he/she completes one lap of the track?</h3>
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<p>Let's look first
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at the route followed by the runner in <strong>Lane 1</strong> (the inside lane).</p>
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<p>The rules state that you measure 0.3 m from the inner edge of the lane (about where the runner runs) for Lane 1 if there is a curb. And 0.2 m for all other lanes:</p>
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<p class="center"><img src="images/athletics-track-200.gif" alt="athletics track 200" height="133" width="360"><br>
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<i>From the IAAF Rulebook, Rule 160.2</i></p>
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<p>So this is what it looks like for Lane 1:</p>
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<p class="center"><img src="images/athletics-track.gif" alt="athletics track diagram" height="446" width="431"></p>
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<p> </p>
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<p>On the curved sections Lane 1 has a radius of 36.5, but we need to <b>add 0.3 m</b> for the "running position", for a total of 36.8 m</p>
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<table width="100%" border="0">
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<tbody>
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<tr>
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<td><img src="images/athletics3.gif" alt="36.8 radius circle" height="190" width="194"></td>
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<td>
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<p>And together the two curved parts make a <b>circle of
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radius 36.8m</b>.</p>
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<div class="words">
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<p>See the page <a href="../geometry/circle.html">Circle</a> to learn more about radius and circumference.</p>
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</div>
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</td>
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</tr>
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</tbody></table>
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<p>So, how far would you have to run? Answer: the Circumference of the circle (plus the straight parts)</p>
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<div class="center80">
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<p>The radius is <strong>36.8 m</strong></p>
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<p>So the Circumference
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= 2 × <span class="times">π</span> × radius = 2 × <span class="times">π</span> × 36.8 m <strong>= 231.22
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m</strong></p>
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<p><b>Add</b> the two straight section of 84.39 m:</p>
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<p class="center">231.22 + 2 × 84.39 m = 231.22 + 168.78 = <b>400 m </b></p>
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</div>
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<p>Wow! The inside lane is exactly 400 m.</p>
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<p>Well, that is how it is designed.</p>
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<h2>But What About Lane 2?</h2>
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<p>Each lane is 1220 wide, so the radius for Lane 2 is 36.5 + 1.22 = <b>37.72 m</b></p>
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<p>And we need to <b>add 0.2 m</b> for the Lane 2 "running position" (remember: 0.3 m for Lane 1, 0.2 m for other lanes), for a total of <b>37.92 m </b></p>
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<div class="center80">
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<p>The radius is <strong>37.92 m</strong></p>
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<p>So the Circumference
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= 2 × <span class="times">π</span> × 37.92 m
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<strong>= 238.26 m</strong>
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(to nearest 0.01m)</p>
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<p>Add the two straight section of 84.39 m:</p>
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<p class="center">238.26
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m + 2 × 84.39 m = 238.26
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m + 168.78 m = <b>407.04 m</b></p>
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</div>
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<p>That is <b>7.04 m longer</b> than Lane 1 ...</p>
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<p class="center">... so Lane 2 should start <b>7.04 m after Lane 1</b> to be fair</p>
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<h2>Your Turn</h2>
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<p>Can you complete the following table?</p>
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td><strong>Lane</strong></td>
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<td width="120"><strong>Radius</strong></td>
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<td width="120"><strong>Circumference</strong></td>
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<td width="120"><strong>Total
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distance </strong></td>
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<td width="120"><strong>Staggered Start</strong></td>
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</tr>
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<tr>
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<td height="40">1</td>
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<td align="center">36.8
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m</td>
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<td align="center">231.22
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m</td>
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<td align="center">400
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m</td>
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<td align="center">0
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m</td>
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</tr>
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<tr>
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<td height="40">2</td>
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<td align="center">37.92
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m</td>
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<td align="center">238.26
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m</td>
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<td align="center">407.04
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m</td>
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<td align="center">7.04
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m</td>
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</tr>
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<tr>
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<td height="40">3</td>
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<td> </td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="40">4</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="40">5</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="40">6</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="40">7</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="40">8</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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</tbody>
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</table>
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<p> </p>
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<p>You should have found that the runner in Lane 8 starts about <b>53
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meters</b> in front of the runner in Lane 1!</p>
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<ul>
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<li>Does that surprise you?</li>
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<li>Is
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it fair?</li>
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</ul>
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<p>It's fair because, with the staggered start, each athlete runs exactly
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400 meters.</p>
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<p>But some people say that
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the athletes in the inner lanes have an advantage because they can see
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the other athletes, and know what work they need to do to catch up.</p>
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<p>On
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the other hand, others argue that the athletes in the outer lanes don't
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have such tight curves to run. So,
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unless all races could be run on a straight stretch (like the 100
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meters), it will never be totally fair.</p>
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<h2>Bonus Activity: Area</h2>
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<p>You might want to investigate the <b>area</b> of each
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lane (imagine you want to paint them different colors).</p>
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<p>The area is made up of the circular area and the straights.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/olympic-area.svg" alt="athletics area Outer minus inner"></p>
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<p>We don't want the athletes running position, we want the <b>radius of
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the edge</b>.</p>
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<p> </p>
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<p>The radius of the inside of
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Lane 1 is <b>36.5 m</b>, so the radius of the outside of Lane 1 (which is
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the
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same as the radius of the inside of Lane 2) must be
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<strong>36.5 m + 1.22 m =
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37.72 m </strong></p>
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<div class="center80">
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<p class="center"><span class="larger">Area = <font face="Times New Roman, Times, serif" size="+2"><b>π</b></font> × radius<sup>2</sup></span> (read more on the <a href="../geometry/circle.html">Circle</a> page)</p>
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</div>
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<p>And the area
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of both straights =
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2 × 1.22 m × 84.39 m = <b>205.9
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m<sup>2</sup></b> (to one decimal).</p>
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<p> </p>
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<h3>You can do the rest! Will the areas be different? By a little, or a lot?</h3>
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td><strong>Lane</strong></td>
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<td align="center" width="90"><strong>Inner
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radius</strong></td>
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<td align="center" width="90"><strong>Outer
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radius</strong></td>
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<td align="center" width="90"><strong>A<sub>In</sub> = Area of circle with Inner radius </strong></td>
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<td align="center" width="90"><strong>A<sub>Out</sub> = Area of circle with Outer radius </strong></td>
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<td align="center" width="90"><strong>A<sub>Out</sub> - A<sub>In</sub></strong></td>
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<td align="center" width="90"><strong>Area
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of both straights</strong></td>
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<td align="center" width="90"><strong>Total
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area of lane</strong></td>
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</tr>
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<tr>
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<td height="30">1</td>
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<td align="center">36.5
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m</td>
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<td align="center">37.72
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m</td>
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<td align="center">4,185.4
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m<sup>2</sup></td>
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<td align="center">4,469.9
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m<sup>2</sup></td>
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<td align="center">284.5 m<sup>2</sup></td>
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<td align="center">205.9
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m<sup>2</sup></td>
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<td align="center">490.4
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m<sup>2</sup></td>
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</tr>
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<tr>
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<td height="30">2</td>
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<td align="center">37.72
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m</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">3</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">4</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">5</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">6</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">7</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">8</td>
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<td><br></td>
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<td><br></td>
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</tr>
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</tbody>
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</table>
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<p> </p>
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