Davros departs on a rocket voyage from the planet Skaro, travelling at speed $u$ (where $0<u<c$ ) in the positive $x$ direction in Skaro's rest frame. After travelling a distance $L$ in Skaro's rest frame, he jumps onto another rocket travelling at speed $v^{\prime}$ (where $0<v^{\prime}<c$ ) in the positive $x$ direction in the first rocket's rest frame. After travelling a further distance $L$ in Skaro's rest frame, he jumps onto a third rocket, travelling at speed $w^{\prime \prime}\left(\right.$ where $0<w^{\prime \prime}<c$ ) in the negative $x$ direction in the second rocket's rest frame.

Let $v$ and $w$ be Davros' speed on the second and third rockets, respectively, in Skaro's rest frame. Show that

$$v=\left(u+v^{\prime}\right)\left(1+\frac{u v^{\prime}}{c^{2}}\right)^{-1}$$

Express $w$ in terms of $u, v^{\prime}, w^{\prime \prime}$ and $c$.

How large must $w^{\prime \prime}$ be, expressed in terms of $u, v^{\prime}$ and $c$, to ensure that Davros eventually returns to Skaro?

Supposing that $w^{\prime \prime}$ satisfies this condition, draw a spacetime diagram illustrating Davros' journey. Label clearly each point where he boards a rocket and the point of his return to Skaro, and give the coordinates of each point in Skaro's rest frame, expressed in terms of $u, v, w, c$ and $L$.

Hence, or otherwise, calculate how much older Davros will be on his return, and how much time will have elapsed on Skaro during his voyage, giving your answers in terms of $u, v, w, c$ and $L$.

[You may neglect any effects due to gravity and any corrections arising from Davros' brief accelerations when getting onto or leaving rockets.