Photon Starship, energy momentum

[George231086]

Hi I'm having trouble with this question and would like some kind of hint on how to proceed.

A photon starship starts from rest and propels itself by emitting photons in the direction opposite to its motion until it reaches a speed v. Use energy momentum conservation law to show that the ratio of the initial rest mass, m, to it's final rest mass M is

m/M = sqrt[(1+v/c)/(1-v/c)].

I'm not sure how to approach it, i think my formulation is wrong. I've looked at:

(Eo/c,0)=(E1/c,p)+kE3/c(1,n) where Eo is the original energy of the spaceship as measured in the initial frame, E1 the energy after and then E3 the energy of the photon. I've added a K to account for how many photons we need to get to a velocity v. However looking at this seems to give nothing. I've also used the identity E^2=E0^2+c^2p^2 to try and get something but it goes nowhere.

Essentially i know that in order to get the final rest mass i need to be in that frame and not the initial frame. I know from the initial frame the mass will be mgamma.

Overall i think my formulation is wrong and that my view of the situation is incorrect and so would like a push in the right direction.

Thanks.

 

[George231086]

Right i think I've got it, if I've fudged it or my reasoning is wrong i'd appreciate a correction.

So I've considered it as if it was a particle which has split into two, a photon and the spaceship in motion. I've taken it from the rest frame of the initial space ship. So using energy-momentum conservation we have that:

4 momentum of spaceship1= 4 momentum of spaceship2 (in motion)+4 momentum of particle.

So (E0/c,0) = (E/c,P) + (Ep/c)(1,n).

Now P=mv=Ep/c and E0=E+Ep, so E0=E+mvc which gives E=E0-mvc.

Then E=moc^2-mvc and we have that mc^2=moc^2-mvc. After rearranging this gives
(m0/m)=1+v/c but m=(gamma)*M so multiplying by gamma we get the result.

Hopefully that's fine. I think my main problem was that i was thinking as the spaceship at the beginning as the same as the one at the end, in the sense that it's rest mass should be the same. Thinking of it as a particle that has exploded seemed to help.