Velocity addition again **
A train of proper length L moves at speed v with respect to the ground. A ball is thrown from the back of the train to the front at speed u with respect to the train. When finding the time of this process in the ground frame, a common error is to use time dilation to go from the train frame to the ground frame, which gives an incorrect answer of γv(L/u). This is incorrect because time dilation is valid only if the two relevant events occur at the same place in one of the frames; otherwise simultaneity becomes an issue.
(a) Find the total time in the ground frame correctly by looking at how much a clock at rest in the train frame advances (for example, a clock at the front of the train), and by applying time dilation to this clock.
(b) Find the total time in the ground frame by applying time dilation to the ball's clock. Your answer will contain the unknown speed V of the ball with respect to the ground.
(c) Equate your results from parts (a) and (b) to show that γV=γuγv(1+uv/c2). Then solve for V to produce the velocity-addition formula.