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.
I just randomly stumbled across this puzzle from yesteryear, and I have to say that I have issues with it. I much prefer the bugs-on-a-stick version of this problem.
ACCELERATION TOWARDS the CENTER. Objects only move in a circle if they are accelerating towards the center. So what is the source of this acceleration, and what happens to it upon collision? If the vehicle is doing the acceleration, because it is powered and its wheels are turned, then it needs to instantaneously reverse the direction and shift the tires. If we are going around a grooved track, and the friction from the outer edge of the groove is the accelerating force, then the cars will eventually stop, perhaps before they are back in their starting position and direction, although this contradicts the premise of the problem. If they are all tethered to the center with a string, I guess that would work, except that the strings are all going to get tangled up.
OTHER FRICTION. Similarly, the cars must be accelerating in the direction of their travel in order to maintain a constant speed despite air friction and rolling friction. So how is this acceleration reversed at the point of impact? Presumably, the author intended that cars with no length also have no friction. But the first objection (acceleration towards the center) still stands.
Stop the race! Trouble on the track!
