Counting Collisions
Five identical balls are placed in a line, spaced equally apart, as shown in the diagram. The balls at both ends are moving with speeds and respectively, towards each other. If all collisions are elastic, how many collisions will take place?
Assume the size of the balls is much smaller than the distance between two adjacent balls.
Hint:
When two balls of equal mass collide elastically, their velocities get exchanged.
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1 solution
The hint is very important: When two balls of equal mass collide elastically, their velocities get exchanged.
Let's go further than that. Since the balls are identical, when two of them collide, instead of saying their velocities get exchanged, we can view it as if they "pass through" each other.
Based on this observation, Since the leftmost ball will "pass through" the 4 balls to its right, and the rightmost ball will "pass through" the 4 balls to its left, there are 4 + 4 − 1 = 7 collisions (we double counted the collision of the leftmost and the rightmost ball).