Davros departs on a rocket voyage from the planet Skaro, travelling at speed (where ) in the positive direction in Skaro's rest frame. After travelling a distance in Skaro's rest frame, he jumps onto another rocket travelling at speed (where ) in the positive direction in the first rocket's rest frame. After travelling a further distance in Skaro's rest frame, he jumps onto a third rocket, travelling at speed where ) in the negative direction in the second rocket's rest frame.
Let and be Davros' speed on the second and third rockets, respectively, in Skaro's rest frame. Show that
Express in terms of and .
How large must be, expressed in terms of and , to ensure that Davros eventually returns to Skaro?
Supposing that 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 and .
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 and .
[You may neglect any effects due to gravity and any corrections arising from Davros' brief accelerations when getting onto or leaving rockets.