How is Rest Mass Affected in the Process of Rocket Propulsion?

[SHawking01]

Homework Statement

A rocket, initially at rest, propels itself in a straight line by giving portions of its mass a constant (backward) speed vout relative to its instantaneous rest frame. (a) Is rest mass conserved in this process? (b) Show that m₀du′ = −dm₀ v_out, where m₀ is the rocket’s rest mass and du′ is the change in the rocket’s speed (in its instantaneous rest frame) when its rest mass changes by dm₀. (c) Show that m₀ du/dm₀ = −(1 − u ²/c²) v_out, where u is the speed of the rocket in its initial rest frame. (d) Suppose the rocket reaches speed u_f relative to its initial rest frame. Show that the ratio of its initial to final rest mass is given by (m_0,i/ m_0,f)= (c + u_f)/( c − u_f)^c/2vout

Homework Equations

Velocity addition formula: u= u'+v/(1+vu'/c²)

The Attempt at a Solution

a)Rest mass is not conserved due to loss of kinetic energy.
b) m₀c²=γ(v_out)d\tilde{\m}c²+ γ(du')(m₀+dm₀)c²
0=γ(du')(m₀+dm₀)du'-γ(v_out)d\tilde{\m}v_out
γ(du')=1/√1-(du')²/c² by the binomial approximation is ≈1.
m₀du'=γ(v_out)d\tilde{\m}v_out
m₀du'=-dm₀v_out
m₀/dm₀=-v_out/du'
c) v_out=m₀du'/-dm₀
I know I have to use the velocity addition formula. How do I find u and u'?

 

[mfb]

Rest mass is not conserved due to loss of kinetic energy.

Where is kinetic energy lost?
How do you define (total?) rest mass in the first place?

For (c), I don't think you need velocity addition, the transformation between u' and u (but I don't understand how exactly they are defined) could give the gamma factor without it.