Question 5

A rocket of initial mass $M_0$ starts from rest and propels itself forwards in the ${\bf e}_1$ direction by emitting photons backwards. The final mass of the rocket after its engine has finished firing is $M_f$. All of the quantities you are asked to find are defined in the frame where the rocket started at rest.

(a) Find the total energy of the exhaust. [Hint: Since all the exhaust is travelling in the same direction at the speed of light, you can treat it as a single ``photon".]

(b) Hence write down the total 4-momentum of the rocket's exhaust.

(c) Find the final 4-momentum of the rocket.

(d) Use your result for the final energy of the rocket to show that its final speed is

$$v={1-y^2\over 1+y^2}c,$$

where $y=M_f/M_0$ is the ratio of the rocket's final and initial masses.

[Hints can be found here, but do not follow this link until you have attempted the question on your own.]