Question

Three train carriages, A, B and C, are on the same track with B between A and C. They all have the same mass, ?, and B is coupled with C. The coupling can be described as a massless spring with the spring constant ?. B and C stand still on the track, while A moves towards B at a constant speed ?0.
A abuts B and undergoes a completely elastic collision. What is the maximum compression of the spring in the clutch?

Previous question
Three train carriages, A, B and C, are on the same track with B between A and C. They all have the same mass, ?, and B is coupled with C. The coupling can be described as a massless spring with the spring constant ?. B and C stand still on the track, while A moves towards B at a constant speed ?0.
A abuts B and undergoes a completely elastic collision. What is the velocity of A (??), B (??), and C (??) at the moment immediately after the collision?

Answer 1

At the moment of the collision, C will not receive any effect of the collision, because the spring will initially compress for some time. Only after the spring has reached the maximum compression will C receive any momentum. So, immediately after the collision

Since A and B have same mass, and since B is at rest, a collision between them will result in an exchange of velocities. This can be shown as follows. Let the velocities of A and B immediately after the collision be and . From conservation of energy,

From conservation of momentum, we get

This can be written in the form

Squaring this equation gives

Subtracting the energy conservation equation from this equation gives

Since A collides with B, B will get some momentum from A and hence its final velocity cannot be 0. So, . That leaves us with the solution

Substituting this in the momentum conservation equation gives

We have all the final velocities. The total kinetic energy at this point is

Since the spring has not started compressing yet, there is no potential energy at this point, so the total energy is just the kinetic energy.

To find the maximum compression, note that at the point of maximum compression, B has just come to rest and C is just about to start moving. Hence, at this point, all the carriages are at rest. The total kinetic energy is 0. All the kinetic energy has been transformed into the potential energy of the spring. Assuming the maximum compression is ,the total energy at this point is

The compression of the spring is a process which conserves energy. Hence, the total energy before and after the compression should be equal, . Substituting the expressions for and gives

This is the expression for the maximum compression of the spring.