Consider the following collision. Frank is at rest in system K holding a ball of mass m. Mary holds a similar ball in system K' that is moving in the x-direction with velocity v with respect to system K. Frank throws his ball along his y-axis, and Mary throws her ball with exactly the same speed along her negative y-axis. The two balls collide in a perfectly elastic collision, and Frank and Mary each catch his or her own ball as it rebounds. Each twin measures the speed of his or her own ball to be U0 both before and after the collision. Initially Mary throws her ball with velocity (primes showing the measurements are in Mary's frame), as follows. W'M = 0 U'M = -Vo My After the elastic collision, the signs on the above expressions are reversed, so the change in momentum as measured by Mary is as follows. Ap' = Now for Frank's ball, we know UE. 0 and Up = U,. The velocity transformations give for Frank's ball as measured by Mary: ty To find y for Frank's ball, note that we have the following. = Then we have the following. 1 V1 - 02/02 Using p' = ymu' along with the reversal of velocities in an elastic collision, we find the following. Ap' = Finally Ap' = Ap' + Ap'm = 0 as required.