Making more tracks

5877 /www/nrich/html/content/id/5877/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <ul id="stemLinks"> <li><a href="http://nrich.maths.org/4872">Warm-up</a></li> <li><a href="http://nrich.maths.org/6010">Try this next</a></li> <li><a href="http://nrich.maths.org/4752">Think higher</a></li> <li><a href="http://nrich.maths.org/1312">Read: mathematics</a></li> <li><a href="http://motivate.maths.org/conferences/conference.php?conf_id=60">Read: science</a></li> <li><a href="http://plus.maths.org/content/maths-plane">Explore further</a></li> </ul> <div> </div> Suppose that the distance between the two wheels on a bike is $1$ unit (note that this is a modelling assumption: see the foot of this problem for more details). The bike moves forward and steers from the front. The rear wheel of the bike traces a curve $y = f(x)$ in the plane for some function $f(x)$. Find an algebraic expression for the path travelled by the front wheel in terms of $x$ and $f(x)$. Further numerical exploration Use a spreadsheet to plot the path of the front and back wheels when the back wheel follows the paths: <ul> <li>$\tan(x)$ between $-1.2$ and $1.2$ radians (like switching from one side of the road to another)</li> <li>$\frac{1}{x}$ between $0.1$ and $4$ (like turning a corner)</li> <li>the arc of a circle (like spiralling down the exit of a multi story carpark)</li> <li>$\sin(x)$ between $0$ and $14$ radians (like weaving through bollards)</li> </ul> Examine the form of these curves. Can you identify any common themes? Can you make any conjectures about the curves? Can you prove any of these conjectures? <div class="framework">NOTES AND BACKGROUND In reality there is not a fixed distance between the points of contact with the ground of the front and back wheels. In a real bicycle the front stem is not vertical and in many bicycles the front wheel is offset from the stem axis so that the point at which the front wheel touches the ground is behind the stem axis extended. As a result the distance between the points of contact of the two wheels varies as the front wheel is turned and the bike tilts. You may wish to experiment with different bicycles to see these effects in action and to consider the effects of different cycling tracks on this. </div></mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Andrei from `Tudor Vianu' National College provided a good <a class="editorial" href="/content/id/5877/DXzipe-Making%20More%20Tracks.doc">solution and investigation</a> of the sorts of curves which are possible in the motion of a bike. Well done Andrei! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> There are a great many situations in which this problem will lead to experience outside the classroom. Asking students to look out for examples of this mathematics in action may lead to some interesting observations. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> We know the coordinates of the rear wheel. We need to find the coordinates of the front wheel. To do this think about the instantaneous direction of travel of the rear wheel of the bike at any moment in time as it travels along its curve. How far away and in what direction is the front wheel located? For the conjecturing, always relate back to the physical experience of riding a bike. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The tangent to the curve at the point of contact of the back wheel is f'(x), the angle to the curve being the inverse tangent of this function. The front wheel is located at a distance 1 along this tangent. The change in the x coordinates is therefore cos(a) and the change in the y coordinate is sin(a), where a is the angle Arctan(f'(x)). so rear is (x, f(x)) means that front is (x+cos(Atan(f'(x)), f (x)+ sin(atan(f'(x))) You can see this in action in the linked spreadsheet (not for release in the problem). The charts of the motion are <mdo:image width="403" height="447" src="invx%20track.JPG" alt=""></mdo:image> <mdo:image width="442" height="515" src="tanx%20track.JPG" alt=""></mdo:image> <mdo:image width="440" height="352" src="sin%20track.JPG" alt=""></mdo:image> The key conjecture is when the rear wheel follows a sin curve then so does the front wheel with an offset of pi/2 since sin(atan(cos)) = sin(atan(cos -pi/2)-pi/2) =cos(atin(sin)) (give or take some minus signs Other conjectures may involve the observation that the curvature of the front path is not less than the curvature of the rear path. </mdoxml> 2 5 0 0 0 0 1 Making more tracks Given the equation for the path followed by the back wheel of a bike, can you solve to find the equation followed by the front wheel? Applications engineering Using, Applying and Reasoning about Mathematics Investigations