Balls bouncing against each other.

Consider a pool table and on it a collection of balls in motion. If we assume there is no friction and no loss of energy during collisions, then it is clear that the balls bounce around forever. But what if we take away the walls? (And assume the table spreads out in every direction endlessly.)

Problem 1: Assuming that all the balls have the same speed, compute an upper bound on the number of collisions; do this for 2 balls, 3 balls, 4 balls.

Problem 2: Arrange a collection of infinitely many balls (moving at the same speed) so that there are an infinite number of collisions.

Problem 3: Assuming that we start with a finite number of balls moving at the same speed, show there are only a finite number of collisions on our wallless pool table. (Hint: If you have n-many balls, reduce to the case that there are (n-1)-many balls by showing that some ball must `leave the group'.)

A complete solution to each of the problems must include a proof; however, partial solutions are always welcome and even encouraged as the problem of the month may have no known solution! Solutions to the Problem of the Month can be submitted in PHSC 423 - the deadline for submission is the last day of November.