A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed \(k\) relative to the spacecraft, in the direction of motion of the spacecraft. At time \(t\), the spacecraft has speed \(v\) and mass \(m\).
- Show, from first principles, that while the spacecraft is ejecting fuel,
$$\frac{dv}{dm} - \frac{k}{m} = 0$$
[5]
At time \(t = 0\), the spacecraft has speed \(U\) and mass \(M\).
- Find the mass of the spacecraft when it comes to rest.
[6]
Given that \(m = Me^{-\alpha t^2}\), where \(\alpha\) is a positive constant, and that the spacecraft comes to rest at time \(t = T\),
- find, in terms of \(U\) and \(T\) only, the distance travelled by the spacecraft in decelerating from speed \(U\) to rest.
[6]