4. 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 \(m\).

1. Show, from first principles, that while the spacecraft is ejecting fuel, $$\frac { \mathrm { d } v } { \mathrm {~d} m } - \frac { k } { m } = 0$$ At time \(t = 0\), the spacecraft has speed \(U\) and mass \(M\).
2. Find the mass of the spacecraft when it comes to rest. Given that \(m = M \mathrm { e } ^ { - \alpha t ^ { 2 } }\), where \(\alpha\) is a positive constant, and that the spacecraft comes to rest at time \(t = T\),
3. find, in terms of \(U\) and \(T\) only, the distance travelled by the spacecraft in decelerating from speed \(U\) to rest.