Suppose that X and Y are two objects with masses and , moving along a line with velocities and . When one of them shunts into the other, they separate, moving at new speeds and . You need two equations to work out the new speeds after the collision.

The first equation is given byconservation of momentum: . If both objects have the same mass (as in the case of these cars) then the M's all cancel, and you are left with the conservation of momentum equation

The second equation is therestitution equation, which says that the coefficient of restitution is equal to . In this problem, the coefficient is 0.75, so the equation becomes

Now, what happens in these car pile-ups? The first collision occurs when car A, travelling at 2m/s, hits car B. If the velocities after this collision are and , then the momentum equation says that . The restitution equation says that That gives you two simultaneous equations for and . Solve them. You should find that and

The next thing that happens is that car B, travelling at 1.75m/s, hits the stationary car C. You can go through the same procedure as for the first collision, solving two simultaneous equations to find the velocities and after the collision. I made them and .

But now car A, still moving at 0.25m/s, is going faster than car B, moving at 0.219m/s. So car A will catch up with car B and there will be another collision. Once again, go through the whole procedure of solving two simultaneous equations to find the velocities and after this collision. You should get and At this stage, each car is going slower than the one in front of it, so there will be no more collisions and the calculation is complete.