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 $n^{(5/2+\varepsilon)n}$, for any fixed $\varepsilon>0$ 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.