Question

2. A spacecraft of initial mass M (including the mass of the fuel) is at rest in the solar system, preparing to disembark on a grand mission to Alpha Centauri. Its engines work by combining matter and antimatter and directing the hard gamma rays that result out the spacecraft’s rear. The engines fire for a brief time, bringing the ship’s speed to v = 0.95 with respect to the solar system. a) What is the rest mass m of the ship after the engines have fired, expressed as a fraction of M? [Hint: Treat the gamma-ray exhaust as a giant photon.] b) Once the ship reaches Alpha Centauri, it must decelerate to a stop by firing its engines again. What will the ship’s total initial mass M have to be if the ship’s empty mass is m0 when it comes to a stop at Alpha Centauri? (Express your result as a multiple of m0, and treat the exhaust when stopping as another giant photon.) c) What would the ratio be in order to make a round trip? This means accelerating to v = 0.95, traveling to Alpha Centauri, coming to a stop, then accelerating back to v = 0.95, travelling back to Earth and finally coming to a stop again, all without refuelling.

Answer 1

(a) During the acceleration phase, the rest mass of the ship reduces. This reduction in mass is converted into the energy of the photon. We use conservation of 4-momentum to solve this problem, by setting c=1.

Let be the 4-momentum of the spaceship at rest, be the 4-momentum when moving with velocity v = 0.95 and be the 4-momentum of the ejected hard gamma rays. Then from conservation of 4-momentum,

where we have used , whereE_v is the energy of the moving spaceship and M' is its rest mass.

solving the quadratic equation for M'

but as therefore

substituting value for velocity,

(b) Similar to above, we first find m₀ in terms of M'.

let be the final 4-momentum of the spaceship at rest with mass m₀. then

thus

thus

(c) to make a round trip, the final mass will be m₀/39. thus the ratio of initial to final mass of the spaceship is 1521. i.e. the initial mass of spaceship M is 1521 times the rest mass of spaceship after a return trip.