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| Bumper Cars (Posted on 2004-12-30) |
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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.
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Submitted by David Shin
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Rating: 2.5000 (4 votes)
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Solution:
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Suppose a ghost is riding in each car. Suppose that when two cars collide, the ghosts in those two cars switch cars.
Note that the ghosts travel in circles without changing directions. Note furthermore that every ghost is always inside some car. Let T be the number of seconds required for each ghost to make a complete circle.
This means that after T seconds, the set of cars will return to its original configuration, meaning that wherever there was a car to start, there is a car now traveling in the same direction.
To complete the proof, note that if each car is where it started, we are done. Otherwise, since the cyclical order of the cars never changes, we know that the current situation is a cyclical permutation of the original one. Repeating will eventually yield the identity permutation. |
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