Speed-time problems at the Olympics

7322 /www/nrich/html/content/id/7322/ 1 2011-06-14T15:01:14 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/7500">Warm-up</a></li> <li><a href="http://nrich.maths.org/7278">Try this next</a></li> <li><a href="http://nrich.maths.org/7109">Think higher</a></li> <li><a href="http://plus.maths.org/content/no-limits-usain">Read: mathematics</a></li> <li><a href="http://nrich.maths.org/2742">Read: science</a></li> <li><a href="http://sport.maths.org/content/leaning-2012">Explore further</a></li> </ul> If you are a teacher, click <a href="http://nrich.maths.org/7322&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on... <div> </div> <table style="width: 600px; height: 933pxborder-spacing:1px;" border="1"> <tbody> <tr> <td style="background-color: rgb(255, 102, 0); text-align: center;padding:1px;"><mdo:image src="Lane1.png"></mdo:image></td> <td> At the 2008 Olympic games, the qualifying standards for the women's 100 metres race was 11.32s. How does this compare with the speed of a bus? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane2.png"></mdo:image></td> <td> At the 2008 Olympics Shelly-Ann Fraser won the women's 100m in a time of 10.78s. If she had continued running, how much further would she have run by the time an athlete running at the qualifying speed (11.32s) would have crossed the line? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%203.png"></mdo:image></td> <td> In the 2009 IAAF World championship, Usain Bolt ran the 100m in 9.58s. Estimate how far he would have been ahead of the gold medallist from Lane 2 had they been racing together. </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%204.png"></mdo:image></td> <td> Imagine that you raced in the 200m with Usain Bolt. By what length would he beat you? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%205.png"></mdo:image></td> <td> Imagine that a 2km rowing race took place on a rowing lake with two separate legs of 1km. How would the total race time vary from a race on a river where one leg is upstream and the other downstream? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%206.png"></mdo:image></td> <td> Imagine that cyclist A completes a lap following the blue line on the <a href="http://www.london2012.com/virtual-tours/velodrome-february-2011.php">velodrome track</a>. Cyclist B completes a lap 1m inside the blue line and cyclist C completes a lap 2m outside the blue line. How do the distances travelled vary between the cyclists? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%207.png"></mdo:image></td> <td> In the past, the start of a 100m race was indicated by a pistol shot next to lane 1. Did this give a significant advantage to the runner in lane 1? Would it have given a significant advantage to anyone if this pistol was fired from the end of lane 4? </td> </tr> <tr> <td style="background-color: rgb(255, 102, 0);padding:1px;"><mdo:image src="Lane%208.png"></mdo:image></td> <td> Imagine an announcement is made from a podium in the centre of a stadium. As the speaker talks into her microphone the sound is simultaneously sent to speakers which project the sound into the stadium and up to satellites which transmit the signal as digital radio. Who might hear the sound first: someone listening on the radio or someone listening in the stadium? </td> </tr> </tbody> </table> Note: In this problem you might not have been given all of the required data and you might need to make estimates or approximations. You should be clear in your mind as to any assumptions that you make whilst constructing your answers. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Lane 1: The required speed to qualify is $\frac{100m}{11.32s}=8.83ms^{-1}.$ This is $8.83\times\frac{3600}{1600}mph = 19.9mph$. This is a similar speed to a bus travelling in a built-up area, but of course the bus will continue at this speed for much longer! Lane 2: Shelly-Ann Fraser travelled at a speed of $\frac{100m}{10.78s}=9.27ms^{-1}.$ If she had continued at this speed until the athlete running at the qualifying speed had finished (an extra $11.32s-10.78s = 0.54s$), she'd have ran an additional $5.01m.$ Lane 3: When Usain Bolt crossed the finish line in $9.58s$, Shelly-Ann Fraser would still have $10.78s-9.58s = 1.2s$ to run, meaning she would be $1.2s\times9.27ms^{-1}=11.1m$ behind. (These calculations assume they run at their average speed throughout the race, which is certainly not the case, as the athletes take several seconds to get to full speed. However, these figures should give an acceptable approximation.) Lane 4: Your 200m time depends on lots of factors including age, gender and fitness, but a good ballpark figure would be 30s. After Usain Bolt finishes in his world record time of 19.19s, you'd still have $\frac{10.81}{30}\times200m=72m$ left to run! (Again, we're assuming constant speed throughout the race.) Lane 5: Suppose on the flat rowing lake, the crew take 210s (3:30) to row 1km. Overall, the time for their 2km race is 420s (7:00). This corresponds to a speed of $4.76ms^{-1}$, which we're assuming is the same in both directions and throughout the race. Suppose on the River Thames, the tide makes the boat go $x ms^{-1}$ faster than before on the downstream section, and $x ms^{-1}$ slower than before on the upstream section. The overall time for this race is therefore $(\frac{1000m}{(4.76-x)ms^{-1}}+\frac{1000m}{(4.76+x)ms^{-1}})s$. If x=1, the overall time on the Thames is 439.3s, 19.3s slower than the race on the lake. However, if x=0.2, the overall time is 392.8s, 27.2s faster than the race on the lake. Lane 6: Velodromes for the Olympics are allowed to measure 250m, 333.3m or 400m in length. We'll assume 400m to simplify the calculations. Assuming half of this distance is along the home and back straights along which there's no difference which line you take, there's 200m around the $180^{\circ}$ turns where the position on the track is important. If we assume the cyclist on the blue line travels 400m per lap, then the radius of the semicircular sections is $\frac{100}{\pi}m =31.8m $. Cyclist B will therefore travel $200m+2\times\pi\times(31.8-1)m=393.7m$ and cyclist C $200m+2\times\pi\times(31.8+2)m=412.6m$. Track cyclists try to cycle as close to the inside of the track as possible in order to minimise the distance travelled. However, if they do this they can end up 'boxed-in' by other cyclists; this is one of the other factors that should be considered. Lane 7: Assuming a lane is 1m across, the sound from the starting pistol only has to travel 8m to reach the athlete in lane 8. This takes approximately $\frac{8m}{340ms^{-1}} = 0.02s$, much less than an average human reaction time, so doesn't give anyone a significant advantage. Firing the gun from the centre of the track by lane 4 would halve the time before the sound has reached everyone, but this isn't necessary. Lane 8: The speed of sound is around $340ms^{-1}$ (amongst other things, it depends on altitude and air temperature). Estimating that the crowd is $100m$ from the podium, the sound will take $\frac{100m}{340ms^{-1}}=0.29s$ to reach them. A geostationary satellite used for communications has an altitude of around 36000km. Radio signals (which move at the speed of light) will take around $2\times\frac{3.6\times10^7m}{3\times10^8ms^{-1}}s = 0.24s$ to travel to and return from the satellite. It seems the two delays are similar, although this calculation neglects any delays in the process of transmitting the signal, either on earth or by the satellite, and also neglects any time spent by the digital radio decoding the signal. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem?</h3> <a href="http://nrich.maths.org/7322">This task</a> provides an engaging context for students to explore speed, distance and time problems. Some of the questions require students to make assumptions or find out extra information. <h3>Possible approach</h3> <a href="/content/id/7322/Speed%20time%20olympics.pdf">This worksheet</a> contains all eight questions and could be distributed to students. <div> </div> <div>Here is a <a href="/content/id/7322/speedtime.ppt">PowerPoint presentation</a> with a separate slide for each question. </div> <div>Here are some ways the questions in this problem could be used:</div> <div> </div> <ol> <li>Display one question at the start of a lesson, give students some time to work out their response, and then discuss as a class different ideas and methods. </li> <li>Give all eight questions out and invite students to work on them in pairs or small groups before bringing the class together to share their answers and debate any disagreements. </li> <li>Give out different questions to different pairs and then invite each pair to present their answer, with the rest of the class acting as critical friends insisting on clear reasoning.</li> </ol> There may be opportunities for cross-curricular links with P.E. where students may have collected their own data about their best times for 100 and 200m. It may be appropriate to adapt some of these questions and use students' own times. It is important to be aware throughout that these questions are (deliberately!) not as 'precisely' stated as typical textbook questions. For example, the phrase 'If she had continued running ...' from lane 2 requires an assumption to be made before computation of an answer. There is no absolutely 'right' way to make these assumptions, although assumptions need to be made clearly. You might need to encourage or reassure the class that they are 'allowed' to make their own sensible assumptions on which to base their calculations if they are unused to working in this way. You might find that rich mathematical discussion emerges from the discussion of the modelling assumptions made on certain parts of the question. <h3>Key questions</h3> What assumptions do you need to make? Is there any extra information you need to know? <h3>Possible extension</h3> The challenging task <a href="/7109">Speedo</a> invites students to think about questions of speed, distance and time where acceleration plays a part. <div> </div> <h3>Possible support</h3> <div> </div> <div>We assume students will use calculators when working on these questions. Inviting students to work together will allow them to support each other. Some of the questions could be worked on as a whole class and solution methods could be modelled on the board. The first four questions are similar in nature, and easier than the last four.</div> <div> </div> <div><a href="/7500">Place Your Orders</a> might provide a suitable introductory activity.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">It is important to be aware throughout that these questions are (deliberately!) not as 'precisely' stated as typical textbook questions. For example, the phrase 'If she had continued running ...' from lane 2 requires an assumption to be made before computation of an answer. There is no absolutely 'right' way to make these assumptions, although assumptions need to be made clearly. One possible assumption might be that a runner runs at their average speed for the full distance.</mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Lane 1: The required speed to qualify is $\frac{100m}{11.32s}=8.83ms^{-1}.$ This is $8.83\times\frac{3600}{1600}mph = 19.9mph$. This is a similar speed to a bus travelling in a built-up area, but of course the bus will continue at this speed for much longer! Lane 2: Shelly-Ann Fraser travelled at a speed of $\frac{100m}{10.78s}=9.27ms^{-1}.$ If she had continued at this speed until the athlete running at the qualifying speed had finished (an extra $11.32s-10.78s = 0.54s$), she'd have ran an additional $5.01m.$ Lane 3: When Usain Bolt crossed the finish line in $9.58s$, Shelly-Ann Fraser would still have $10.78s-9.58s = 1.2s$ to run, meaning she would be $1.2s\times9.27ms^{-1}=11.1m$ behind. (These calculations assume they run at their average speed throughout the race, which is certainly not the case, as the athletes take several seconds to get to full speed. However, these figures should give an acceptable approximation.) Lane 4: Your 200m time depends on lots of factors including age, gender and fitness, but a good ballpark figure would be 30s. After Usain Bolt finishes in his world record time of 19.19s, you'd still have $\frac{10.81}{30}\times200m=72m$ left to run! (Again, we're assuming constant speed throughout the race.) Lane 5: Suppose on the flat rowing lake, the crew take 210s (3:30) to row 1km. Overall, the time for their 2km race is 420s (7:00). This corresponds to a speed of $4.76ms^{-1}$, which we're assuming is the same in both directions and throughout the race. Suppose on the River Thames, the tide makes the boat go $x ms^{-1}$ faster than before on the downstream section, and $x ms^{-1}$ slower than before on the upstream section. The overall time for this race is therefore $(\frac{1000m}{(4.76-x)ms^{-1}}+\frac{1000m}{(4.76+x)ms^{-1}})s$. If x=1, the overall time on the Thames is 439.3s, 19.3s slower than the race on the lake. However, if x=0.2, the overall time is 392.8s, 27.2s faster than the race on the lake. Lane 6: Velodromes for the Olympics are allowed to measure 250m, 333.3m or 400m in length. We'll assume 400m to simplify the calculations. Assuming half of this distance is along the home and back straights along which there's no difference which line you take, there's 200m around the $180^{\circ}$ turns where the position on the track is important. If we assume the cyclist on the blue line travels 400m per lap, then the radius of the semicircular sections is $\frac{100}{\pi}m =31.8m $. Cyclist B will therefore travel $200m+2\times\pi\times(31.8-1)m=393.7m$ and cyclist C $200m+2\times\pi\times(31.8+2)m=412.6m$. Track cyclists try to cycle as close to the inside of the track as possible in order to minimise the distance travelled. However, if they do this they can end up 'boxed-in' by other cyclists; this is one of the other factors that should be considered. Lane 7: Assuming a lane is 1m across, the sound from the starting pistol only has to travel 8m to reach the athlete in lane 8. This takes approximately $\frac{8m}{340ms^{-1}} = 0.02s$, much less than an average human reaction time, so doesn't give anyone a significant advantage. Firing the gun from the centre of the track by lane 4 would halve the time before the sound has reached everyone, but this isn't necessary. Lane 8: The speed of sound is around $340ms^{-1}$ (amongst other things, it depends on altitude and air temperature). Estimating that the crowd is $100m$ from the podium, the sound will take $\frac{100m}{340ms^{-1}}=0.29s$ to reach them. A geostationary satellite used for communications has an altitude of around 36000km. Radio signals (which move at the speed of light) will take around $2\times\frac{3.6\times10^7m}{3\times10^8ms^{-1}}s = 0.24s$ to travel to and return from the satellite. It seems the two delays are similar, although this calculation neglects any delays in the process of transmitting the signal, either on earth or by the satellite, and also neglects any time spent by the digital radio decoding the signal. </mdoxml> 2 3 0 0 0 1 0 Speed-time problems at the Olympics Have you ever wondered what it would be like to race against Usain Bolt? Applications Maths Supporting SET Measures and Mensuration Length/distance Measures and Mensuration Speed Applications sport Applications STEM - General Applications STEM - physical world Applications STEM - living world Admin Discussion Secondary Mapping Document Units of measurement Secondary Mapping Document DisplayCabinet STEM mapping STEM - Distance-Speed-Time STEM mapping STEM - Real World Stage 5 Mechanics mapping document Kinematics M1 STEM mapping STEM - maths supporting SET