On the number of hard ball collisions

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are n balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than n − n , then the total number of collisions is bounded by n ( 5 / 2 + ε ) n , for any fixed ε > 0 and large n . We also show that if there is a number of collisions larger than n c nfor an appropriate c > 0 , then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.