# Total number of collisions

Given $$n$$ point masses on the real axis with their initial positions and velocities, determine the total number of collisions (from $$t=0$$ to $$t=\infty$$). Here I suppose that the collisions are elastic, i.e the momentum and energy of 2 particles just before and after each collision are conserved.

For example, if all particles have the same mass, then after each collision, 2 particles exchange their velocities, so we can imagine 2 particles pass through each other. Let $$\sigma_0 = id$$ and $$\sigma_T$$ the order of particles for $$t=0$$ and $$t=T$$, where $$T$$ sufficiently large so that there is no collision for $$t>T$$. The the number of collision is the inversion number of $$\sigma_T$$

What about the general case? What if we replace points by segments? Could you give me some results one has obtained so far?