We derived the differential equation of motion of a nonrelativistic rocket, by conserving both momentum and total mass over a short time interval \(\Delta t\). That is, the momentum of the rocket at time \(t\) was set equal to the sum of the momenta of the rocket and bit of exhaust at time \(t+\Delta t\), and similarly the mass of the rocket at \(t\) was set equal to the sum of the masses of the rocket and bit of exhaust at time \(t+\Delta t\). We can find the equation of motion of a relativistic rocket in a similar way, except that the total mass is not conserved in this case; it is now the total momentum and the total energy that are conserved. At time \(t\), let the rocket have mass \(m\) and velocity \(v\); and at time \(t+\Delta t\) let the rocket have mass \(m+\Delta m\) (where \(\Delta m<0\) ) and velocity \(v+\Delta v\), and let the bit of exhaust have mass \(\Delta M\) and backward velocity \(\bar{u}\). Note that \(\Delta M eq-\Delta m\) in relativistic physics. (a) Show from the velocity transformation that the velocity \(u\) of the bit of exhaust in the instantaneous rest frame of the rocket is given by

\[u=\frac{\bar{u}+v}{1+\bar{u} v / c^{2}}\]

(b) By conserving momentum show that, to first order in changes in \(m, v\), and \(M\),

\[\frac{\Delta m v}{\sqrt{1-v^{2} / c^{2}}}+\frac{m \Delta v}{\left(1-v^{2} / c^{2}\right)^{3 / 2}}=\frac{\Delta M \bar{u}}{\sqrt{1-\bar{u}^{2} / c^{2}}}\]

(c) Then conserve energy, again keeping no terms beyond those with first order changes. Using both the conservation of momentum and conservation of energy expressions, show that the terms with \(\Delta M\) can be eliminated. Then the results of problem (a) can be used to eliminate \(\bar{u}\) in favor of \(u\), and by dividing through by \(\Delta m\) and taking the limit \(\Delta m \rightarrow 0\) show that

\[m \frac{d v}{d m}+u\left(1-\left(v^{2} / c^{2}\right)\right)=0\]

which is the relativistic rocket differential equation of motion. Show also that this reduces to the equation for a classical rocket in the limit of small velocities.