The N Points and N+1 Photos

Fix a number N. Place N indistinguishable points on the number line. Each point has some fixed velocity. A snapshot of the line gives you the positiions and multiplicity of points at certain locations for a fixed time value. That is, if two or more points are at the same location, you are given the number of points there. Prove that if you are given N+1 snapshots of the points at distinct moments in time you can find the velocity of each of the points.

Kell and I have a pretty interesting solution, but it doesn't seem to be as nice of a solution as the ant problem. Check back in a week.