If I have $N$ particles which move in one dimensions, and that collide elastically, is there a way to solve for their trajectories? There are $N$ particles on a line, and I know each of their masses and initial velocities and positions, and that the total energy and momentum is conserved when they collide.
Is there a way to solve for their positions as a function of time exactly? Or is there a way to do it statistically? I'm very interested to find out.
 A: It depends on how realistic and how precise you want your model to be.
The first and most obvious simplification is to work in a reference frame in which the centre of mass of the particles is at rest.
After making that simplification, the very simplest model is to model each particle as a point mass, and assume each collision takes zero time.
If all of the particle masses are equal, this model allows a further elegant simplification. When two point particles with equal masses collide elastically they simply exchange velocities, so you can actually model the particles as passing straight through each other and dispense with the collisions.
However, if the particle masses are not equal this simplification does not work. You will have to calculate the time of the first collision and the velocities of the particles after that collision, and then go on to find the time of the second collision etc. This can become complex if you have a large number of particles, but it can be handled with a fairly straight forward computer simulation.
If you want a more realistic model then you could give each particle a size., and take this into account when modelling each collision. Collisions will still take zero time if you assume the particles are rigid.
An even more realistic model would treat each particle as a small spring which contracts in each collision, and then expands again. Collisions now take non-zero time, and you have to take account of the deformation of each particle during each collision.
