Imagine a particle moving right at 10 mph. It enters a cylinder with an open left end and closed right end. The cylinder is moving right at 1 mph. In the frame of reference of the cylinder, the particle is moving right at 9 mph.
Assume a 100% elastic collision with the closed right end of the cylinder. The particle is now moving left at 9 mph with respect to the cylinder frame and left at 8 mph with respect to the original frame. Part of the momentum of the particle must have been passed to the cylinder.
From the perspective of someone inside the cylinder and moving along with it the particle collided with the end of the cylinder and rebounded without moving the cylinder wall. It looked like the cylinder wall was stationary. This assumes that the observer is somehow attached to the cylinder so that the cylinder itself never appears to move.
From the perspective of someone outside the cylinder even though the cylinder may be opaque the outside observer could tell when the collision occurred because the cylinder's velocity to the right increased at that moment. This must be so because the overall momentum of the particle and the cylinder in the original frame must be preserved. Since the particle reversed course, the cylinder must increase its speed to the right to compensate for the change in the particle's momentum.
To the observer inside and attached to the cylinder, she could tell when the collision occurred even without watching because she would briefly feel a force pushing her to the right when the particle strikes the cylinder--similar to the force one feels when an elevator starts moving upward.
Is this right so far?
If so, this also implies that when a particle strikes a fixed wall and rebounds with 100% elasticity, the wall must move to the right. If the reason the wall is fixed is that it is anchored to some larger structure, the larger structure must move to the right. In other words, since the total momentum of the particle and object it strikes must be constant, when the particle rebounds the object being struck must increase its rightward momentum. In other words, there really is no such thing as a fixed wall from which a particle can rebound without transferring some momentum to the wall.
This all sounds simple, but it grew out of some other question, and I'm here trying to explain how I now understand the larger issue. I had phrased my other question as if there can be a fixed wall from which a particle can rebound without affecting the wall. That was a wrong assumption and lead me down some wrong paths.