Past exam of the physics 1 paper of the University of Oxford / 2015 / 6 / Solution

Align both frames' $x$ axes with the motion and make their origins coincide at $t=t^{\prime }=0$. With $S^{\prime }$ moving along $+x$ at speed $v_2$ relative to $S$, the Lorentz transformation and its inverse arewhere ${\gamma }_2=(1-v_2^2/c^2{)}^{-1/2}$; the transverse coordinates are unchanged. Along the object's path, $d{x}^{\prime }/d{t}^{\prime }=v_1$. Taking differentials of the inverse transformation givesThis is relativistic velocity addition. For $v_1,v_2\ll c$ it reduces to the Galilean sum; if $v_1=c$, the result is still $c$. Two subluminal positive speeds give a subluminal result.