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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

[tex]\frac{v}{c}[/tex]=[tex]\frac{Mi^2-Mf^2}{Mi^2+Mf^2}[/tex]

## 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:

Before:

Rocket is stationary. Therefore

P(before)=Mi*v=0

E(before)=Mi*c^2

After:

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