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SHawking01
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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 vout, 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) vout, where u is the speed of the rocket in its initial rest frame. (d) Suppose the rocket reaches speed uf relative to its initial rest frame. Show that the ratio of its initial to final rest mass is given by (m0,i/ m0,f)= (c + uf)/( c − uf)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(vout)d\tilde{\m}c2+ γ(du')(m0+dm0)c2
0=γ(du')(m0+dm0)du'-γ(vout)d\tilde{\m}vout
γ(du')=1/√1-(du')2/c2 by the binomial approximation is ≈1.
m0du'=γ(vout)d\tilde{\m}vout
m0du'=-dm0vout
m0/dm0=-vout/du'
c) vout=m0du'/-dm0
I know I have to use the velocity addition formula. How do I find u and u'?