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ajclarke
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Homework Statement
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
\frac{v}{c}=\frac{Mi^2-Mf^2}{Mi^2+Mf^2}
Homework Equations
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.
Before:
Rocket is stationary. Therefore
P(before)=Mi*v=0
E(before)=Mi*c^2
After:
E(photons)=pc
p(photons)=E/c
E(after)=Gamma*Mf*c^2
[(after)=Gamma*Mf*v
Calculations
1. Conservation of E
E(before)=E(photons)+E(after)
Mi*c^2=E{photons)+Gamma*Mf*c^2
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:
Mi^2*Mf^2=Beta(Mf^2*Beta+2Mf^2+Mi^2*Beta)
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.
Adam
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