5. A space-ship is moving in a straight line in deep space and needs to reduce its speed from \(U\) to \(V\). This is done by ejecting fuel from the front of the space-ship at a constant speed \(k\) relative to the space-ship. When the speed of the space-ship is \(v\), its mass is \(m\).

1. Show that, while the space-ship is ejecting fuel, \(\frac { \mathrm { d } m } { \mathrm {~d} v } = \frac { m } { k }\). The initial mass of the space-ship is \(M\).
2. Find, in terms of \(U , V , k\) and \(M\), the amount of fuel which needs to be used to reduce the speed of the space-ship from \(U\) to \(V\).
   (6)