A ball moves at speed v1 with respect to a train. The train moves at speed v2 with respect to the ground. What is the speed of the ball with respect to the ground? Solve this problem (that is, derive the velocity addition formula, equation. (10.28)) in the following way. (Do not use time dilation, length contraction, etc. Use only the fact that the speed of light is the same in any inertial frame.) Let the ball be thrown from the back of the train. At this instant, a photon is released next to it (see Figure). The photon heads to the front of the train, bounces off a mirror, heads back, and eventually runs into the ball. In both frames, find the fraction of the way along the train the meeting occurs, and then equate thesefractions.

Ar' + vAt At + v2Ar' /c Aa' /At' + v2 1+(r'/e vi + v2 1+ vjv2/c Ar At (10.28) V2