 Solution

If the primed frame moves at speed $v$ in the positive $x$ direction relative to the unprimed frame, the one-dimensional Lorentz transformation isFor a trajectory with velocity $q=dx/dt$, differentiating these expressions gives $q^{\prime }=(q-v)/(1-vq/c^2)$. Solving for $q$, when the particle has primed-frame speed $u$, proves the relativistic velocity-addition formulaFor the train problem the sign of the frame velocity reverses: the train frame $S$ moves at $+v$ relative to the station frame $S^{\prime }$. Put $\gamma =(1-v^2/c^2{)}^{-1/2}$. The transformation appropriate to the subsequent parts is therefore