M5 June 2013 (R) Q3
3. A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of 2000 m s\(^{-1}\) relative to the spacecraft. The burnt fuel is ejected at a constant rate of \(c\) kg s\(^{-1}\). At time \(t\) seconds the total mass of the spacecraft, including fuel, is \(m\) kg and the speed of the spacecraft is \(v\) m s\(^{-1}\).
(a) Show that, while the spacecraft is ejecting burnt fuel,\[m\frac{\mathrm{d}v}{\mathrm{d}t} = 2000c\] (7)
At time \(t = 0\), the mass of the spacecraft is \(M_0\) kg and the speed of the spacecraft is 2000 m s\(^{-1}\). When \(t = 50\), the spacecraft is still ejecting burnt fuel and its speed is 6000 m s\(^{-1}\).
(b) Find \(c\) in terms of \(M_0\). (7)
| Scheme | Marks |
|---|---|
| \((m + \delta m)(v + \delta v) - (-\delta m)(2000 - v) = mv\) | M1 A2 |
| \(mv + v\delta m + m\delta v + 2000\delta m - v\delta m = mv\) | |
| \(m\delta v = -2000\delta m\) | A1 |
| \(m\dfrac{\mathrm{d}v}{\mathrm{d}t} = -2000\dfrac{\mathrm{d}m}{\mathrm{d}t} = -2000 \times -c = 2000c\) * | M1 M1 A1 |
| (7) |
| Scheme | Marks |
|---|---|
| \(m = M_0 - ct\) | B1 |
| \(\displaystyle\int_{2000}^{6000}\mathrm{d}v = 2000\int_0^{50}\frac{c}{M_0 - ct}\,\mathrm{d}t\) | M1 A1 |
| \(\left[v\right]_{2000}^{6000} = -2000\left[\ln(M_0 - ct)\right]_0^{50}\) | M1 A1 |
| \(2 = \ln\left(\dfrac{M_0}{M_0 - 50c}\right)\) | M1 |
| \(c = \dfrac{M_0}{50}\left(1 - \dfrac{1}{e^2}\right)\) | A1 |
| (7) | |
| (14 marks) |