Some bumper cars are moving around a circular track at the same constant speed. However, they are not all going in the same direction. Collisions are perfectly elastic, so that two colliding cars instantaneously change directions (and continue at the same speed).
Show that at some point in the future, all the cars will be back to their starting positions and directions. Assume that each car has no length.
Imagine that instead of elastic collisions, the cars passed right through each other to continue their journey. Since they are all going at the same speed, one revolution later each car would be at its original starting point going the same direction (since in this altered problem the cars never change direction).
So clearly there will be infinite times, at one period intervals where the above conditions will be true.
But with collisions, (the problem as described) we don't know which car will be where. We do know that the order of the cars will always be the same, since they never pass each other. So once N periods have gone by, there should be at least one time when each car was at its original starting location.

Posted by Larry
on 20041230 18:12:07 