What Happens When a Ball Hits a Moving Wall?

[Domnu]

Problem
If you throw a ball with velocity $$v$$ at a wall, it rebounds with the same speed, $$v$$. What happens if you throw it at speed $$v$$ towards a wall which is traveling towards you at speed $$w$$? What is your answer in the limit in which $$w$$ is much larger than $$v$$?

Answer
Let us work in the frame of the wall. The ball comes towards you at velocity $$v+w$$, so it must leave at velocity $$v+w$$ with respect to you. Moving back to the lab frame, this means that the ball moves now, to the left with velocity $$v+2w$$. In the limit where $$w$$ is much larger than $$v$$, the velocity of the ball is $$2w$$.

Is my answer correct?

 

[Doc Al]

Looks good to me!

 

[Moetasim]

Yes, your answer is correct. In the limit where w is much larger than v, the velocity of the ball after rebounding from the wall will approach 2w, as observed from the lab frame. This is because the wall is moving towards the ball at a much higher speed, which adds to the ball's initial velocity and results in a larger rebound velocity. This can also be explained by the conservation of momentum, where the total momentum of the ball and wall system remains constant before and after the collision.