M5 June 2011 Q3
3. A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass \(M\), of which a mass \(kM\), \(k < 1\), is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed \(c\), relative to the rocket, in a direction opposite to that of the rocket’s motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt. (7)
| Scheme | Marks |
|---|---|
| \((m + \delta m)(v + \delta v) + (-\delta m)(v - c) = mv\) \(m\delta v + c\delta m = 0\) | M1A2 |
| \(\displaystyle\int_0^V \mathrm{d}v = -c\int_M^{M(1-k)} \dfrac{\mathrm{d}m}{m}\) | M1A1 |
| \(V = c[\ln m]_{M(1-k)}^{M}\) | A1 |
| \(V = c\ln\left(\dfrac{1}{1-k}\right)\) | A1 |
| (7 marks) |