Paper 4, Section II, B

(a) A rocket, moving non-relativistically, has speed $v(t)$ and mass $m(t)$ at a time $t$ after it was fired. It ejects mass with constant speed $u$ relative to the rocket. Let the total momentum, at time $t$, of the system (rocket and ejected mass) in the direction of the motion of the rocket be $P(t)$. Explain carefully why $P(t)$ can be written in the form

$\tag{*} P(t)=m(t) v(t)-\int_{0}^{t}(v(\tau)-u) \frac{d m(\tau)}{d \tau} d \tau$

If the rocket experiences no external force, show that

$\tag{†} m \frac{d v}{d t}+u \frac{d m}{d t}=0$

Derive the expression corresponding to $(*)$ for the total kinetic energy of the system at time $t$. Show that kinetic energy is not necessarily conserved.

(b) Explain carefully how $(*)$ should be modified for a rocket moving relativistically, given that there are no external forces. Deduce that

$\frac{d(m \gamma v)}{d t}=\left(\frac{v-u}{1-u v / c^{2}}\right) \frac{d(m \gamma)}{d t}$

where $\gamma=\left(1-v^{2} / c^{2}\right)^{-\frac{1}{2}}$ and hence that

$\tag{‡} m \gamma^{2} \frac{d v}{d t}+u \frac{d m}{d t}=0$

(c) Show that $(†)$ and $(‡)$ agree in the limit $c \rightarrow \infty$. Briefly explain the fact that kinetic energy is not conserved for the non-relativistic rocket, but relativistic energy is conserved for the relativistic rocket.

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