Making Tracks
Problem
A bicycle passes along a path and leaves some tracks.
Is it possible to examine the curve of the individual tracks (both tyres have identical tread) and from them say which track was made by the front wheel and which by the back wheel?
Is it possible to say in which direction along the path the bicycle was travelling ?
The image above is just a picture to suggest the context, but if you'd like some accurate curves (tracks) to play with see the HintGetting Started
Perhaps find a bicycle and experiment.
Or perhaps trace a curve on paper and imagine a cyclist following that curve with the front wheel, and then consider the possibilities for the back wheel ?
You may also like to consider the motions recorded in these three images. One line traces the path of a back wheel and one line traces the path of a front wheel:
Student Solutions
Jack from Wolgarston High School gave this a lot of thought, here are some more ideas about the bike tracks :
We know that there's an important difference between a back wheel and a front wheel : the front wheel can point in any direction, just turn the handle bar, but the back wheel always points exactly forward towards the front wheel, or more exactly, towards the point where the front wheel has contact with the ground.
So we could test to see if a track could have been made by that back wheel :
From the track being tested draw out a line that seems to be the wheel's direction at that point, and then continue that direction line until it hits the other track - that's where the front wheel would have needed to be.
Do that for lots of points along the 'tested back wheel track', and if that was a genuine back wheel track the distance along any of those direction lines to the front wheel contact point would be a constant length.
That's because, although the front wheel can turn the contact point with the ground is the same however you angle the handle bar.
If the line first tested is unsuccessful the other track ought to be the back wheel, but it's a good idea to do the test and check !
The straight line from a point on the back wheel track which shows the direction of the back wheel at that point is called the 'tangent'.
Teachers' Resources
For most students this problem will be outside their ordinary mathematical experience.
There is no frequently-used technique which may be just applied in this new situation.
Talk helps enormously in a problem like this. Questions to prompt thinking include :
- What difference can you think of between the front and the rear wheels ?
- What is the connection between the front and rear wheels which, for example, wouldn't be possessed by two uni-cyclists going along the path ?
These suggestions from the hint page may be useful :
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Find a bicycle and experiment.
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Trace a curve on paper and imagine a cyclist following that curve with the front wheel, and then consider the possibilities for the back wheel ?
Above all this is a good problem not to rush .
Leave the way into this problem left open and allow students to bring along fresh thoughts from time to time. The emerging solution will be much more satisfying and the process a much better experience of real new mathematics emerging.