Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 8B / Solution

Take $S^{\prime }$ to move along $+x$ at velocity $V$, with the origins coincident at zero time. The Lorentz transformation isDifferentiating along the particle trajectory gives the relativistic velocity-addition formulaFor a timelike particle, $|u|<c$ and $|V|<c$. Direct algebra givesso $|u^{\prime }|<c$, in particular $u^{\prime }<c$. Strictness requires a massive, subluminal particle; for a photon with $u=\pm c$, the transformation preserves $u^{\prime }=\pm c$ instead.