A rocket of mass $m(t)$, which includes the mass of its fuel and everything on board, moves through free space in a straight line at speed $v(t)$. When its engines are operational, they burn fuel at a constant mass rate $\alpha$ and eject the waste gases behind the rocket at a constant speed $u$ relative to the rocket. Obtain the rocket equation

$$m \frac{d v}{d t}-\alpha u=0$$

The rocket is initially at rest in a cloud of space dust which is also at rest. The engines are started and, as the rocket travels through the cloud, it collects dust which it stores on board for research purposes. The mass of dust collected in a time $\delta t$ is given by $\beta \delta x$, where $\delta x$ is the distance travelled in that time and $\beta$ is a constant. Obtain the new equations

$$\begin{aligned} \frac{d m}{d t} &=\beta v-\alpha \\ m \frac{d v}{d t} &=\alpha u-\beta v^{2} \end{aligned}$$

By eliminating $t$, or otherwise, obtain the relationship

$$m=\lambda m_{0} u \sqrt{\frac{(\lambda u-v)^{\lambda-1}}{(\lambda u+v)^{\lambda+1}}},$$

where $m_{0}$ is the initial mass of the rocket and $\lambda=\sqrt{\alpha / \beta u}$.

If $\lambda>1$, show that the fuel will be exhausted before the speed of the rocket can reach $\lambda u$. Comment on the case when $\lambda<1$, giving a physical interpretation of your answer.