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^cn for an appropriate c > 0, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.