2. Relativistic matter-antimatter rocket. Assume that a rocket propels vertically away from the Earth's surface itself by continually converting some of its mass into photons and firing them out the back. The rocket starts from rest, and has initial mass M. Let m be its instantaneous mass, and be the rate of which mass is consumed in the rocket's instantaneous rest frame. Let u be its instantaneous speed with respect to the ground. Ignore the force due to gravity.

(a) In the instantaneous rest frame, S, of the rocket, what is the momentum of the photons generated in time dt? State clearly the physical principles you are using.

(b) In S, what is the momentum imparted to the rocket in dt?

(c) What is the force on the rocket in S?

(d) Using the force transformations, determine the force on the rocket in S.

(e) Write down the equation of motion of the rocket in S.

(f) From this equation of motion, show that dm/m+du/c(1u2/c2) = 0. NB: in manipulating the equation of motion (specifically, while differentiating the rocket's momentum with respect to time), you should treat m as a constant.

(g) Integrate the above relation to obtain m as a function of u.

(h) Combine the above relation with that the rate at which the mass is consumed to obtain a differential equation in du/dt. Integrate the equation to obtain t as a function of u, and work out the asymptotic form of u as a function of t in the case where u = c(1 ), with 1.