AP Physics C: Does the release of a spring add momentum to a closed system?

Question

Two carts of equal mass are on a horizontal surface with negligible friction. Cart A is approaching cart B, which is at rest. Attached to cart B is a spring that is initially compressed. At the moment cart A collides with cart B, the spring is released and pushes on cart A.

Which of the following correctly states what happens to the kinetic energy and the momentum of the two-cart system as a result of the collision compared to those quantities before the collision?

The answer according to this AP Physics exam is that Kinetic Energy increases and the magnitude of momentum stays the same.

Why does the magnitude of momentum stay the same? I would expect momentum to increase since the spring added force to the system.

Answer 1

The net momentum of a system (the sum of the momenta of its parts) can never be increased unless a net force is applied to the system as a whole. Every force has an equal and opposite reaction, so if all forces only come from within the system, the change in momentum $\Delta p = \mathbf{F}\Delta t$ resulting from a force is cancelled by $-\Delta p = -\mathbf{F}\Delta t$ resulting from the opposite force of the action/reaction pair, which also is within the system.

Answer 2

Think of this experiment in a slightly different context. If the same two carts started at rest, touching each other, and the spring in one of the carts was activated, what would happen? The system starts with zero total momentum with the spring having stored elastic potential energy. Once the spring is activated, the spring causes the carts to move away from each other. Since each cart is now moving, the sum of the kinetic energy of the two carts is now positive and equal to the original spring potential energy. However, it should be obvious that each cart has a momentum that is equal in magnitude to the momentum of the other cart, but opposite in sign. Thus, the starting total momentum of the two carts is zero, and the ending momentum of the two carts is also zero. The same starting vs. ending momentum of the two carts applies if the carts are initially moving towards each other.

Answer 3

In the case of this problem, the system is composed of cart A, cart B and the spring attached to cart B. It is closed because no external forces are acting on the system (or on any of its components) and there are is no exchange of matter/energy with the environment. It is assumed that cart A has some initial velocity (assumed to have come from infinity with that velocity relative to cart B).

Why does the magnitude of momentum stay the same?

The total momentum is constant (conserved) for a closed system.

I would expect momentum to increase since the spring added force to the system.

I think the spring is supposed to be considered as part of cart B, so that the spring force is an internal force of the system. Thus, the spring did not "add force" to the system. The same thing could be said of having repelling magnets on the carts, or even if the carts were made of rubber and they actually collide and bounce.