A spacecraft begins a journey with rest mass Mi. Its method of propulsion involves converting matter entirely to photons, which are emitted in the direction opposite to the direction of motion. After a period of acceleration the rest mass has been reduced to Mf. Show that the velocity v of the spacecraft, relative to its initial rest-frame, is then given by
Conservation of Momentum
Conservation of Energy
The Attempt at a Solution
Well I think I have at least got the Physics right to this.
Thining in two stages.
Rocket is stationary. Therefore
1. Conservation of E
2. Conservation of P
P(before)=P(after)-P(photons) [Negative for photons since ejected in -ve x direction]
0=Gamma*Mf*v - E(photons)/c
Rearranging this: E(photons)=Gamma*Mf*v*c
I then subbed this in for consv. of E equation above. Multiplied by the denomintor of Gamma, squared out to get rid of the root etc, and at a suggestion from the lecturer let Beta=v/c
However when I square all the terms, I end up with a factor of 2 in it. Bascially I end up with:
Which clearly will not rearrange to the answer. I have worked through this about 5 or six times now all to no avail. Anyone able to see a mistake or omething I have missed out. I checked the Physics at the beginning with the lecturer and he said it was fine, so i think its some mathmatical treatment which is causing me issues here.
Once again, thank you for your help.