A train with proper length L moves at speed c/2 with respect to the ground. A ball is thrown from the back to the front, at speed c/3 with respect to the train. How much time does this take, and what distance does the ball cover, in:

(a) The train frame?

(b) The ground frame? Solve this by

 i. Using a velocity-addition argument.

 ii. Using the Lorentz transformations to go from the train frame to the ground frame.

(c) The ball frame?

(d) Verify that the invariant interval is indeed the same in all three frames.

(e) Show that the times in the ball frame and ground frame are related by the relevant °-factor.

(f) Ditto for the ball frame and train frame.

(g) Show that the times in the train frame and ground frame are not related by the relevant γ-factor. Why not?