M5 June 2016 Q6
6. A firework rocket, excluding its fuel, has mass \(m_0\) kg. The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed \(u\) m s\(^{-1}\), \(u \gt 24.5\), relative to the rocket. The rocket starts from rest on the ground at time \(t = 0\). At time \(t\) seconds, \(t \leqslant 2\), the speed of the rocket is \(v\) m s\(^{-1}\) and the mass of the rocket including its fuel is \(m_0(5 - 2t)\) kg. It is assumed that air resistance is negligible and the acceleration due to gravity is constant.
| Scheme | Marks |
|---|---|
| \(-mg\delta t = (m + \delta m)(v + \delta v) + (-\delta m)(v - u) - mv\) | M1 A2 |
| \(-mg\delta t = m\delta v + u\delta m\) | |
| \(-g = \dfrac{\mathrm{d}v}{\mathrm{d}t} + \dfrac{u}{m}\dfrac{\mathrm{d}m}{\mathrm{d}t}\) | |
| \(m = m_0(5 - 2t) \Rightarrow \dfrac{\mathrm{d}m}{\mathrm{d}t} = -2m_0\) | B1 |
| \(-9.8 - \dfrac{u}{m_0(5 - 2t)} \cdot (-2m_0) = \dfrac{\mathrm{d}v}{\mathrm{d}t}\) | M1 |
| \(\dfrac{2u}{(5 - 2t)} - 9.8 = \dfrac{\mathrm{d}v}{\mathrm{d}t}\) PRINTED | A1 |
| (6) |
Notes
First M1 for a general impulse-momentum equation (correct number of terms, excluding any \(\delta m\delta v\) terms)
First A2 for a correct equation
B1 for \(m = m_0(5 - 2t) \Rightarrow \dfrac{\mathrm{d}m}{\mathrm{d}t} = -2m_0\)
Third M1 for substituting for \(\mathrm{d}m/\mathrm{d}t\) to produce an equation in \(v\) and \(t\) only.
Third A1 for PRINTED ANSWER (need 9.8 not \(g\))
| Scheme | Marks |
|---|---|
| \(v = -u\ln(5 - 2t) - 9.8t\ (+C)\) | M1 A1 |
| \(t = 0,\ v = 0 \Rightarrow C = u\ln 5\) | M1 |
| \(v = u\ln\dfrac{5}{(5 - 2t)} - 9.8t\) | A1 |
| \(m_0 = m_0(5 - 2t) \Rightarrow t = 2\) | M1 |
| \(v = u\ln 5 - 19.6\) (m s\(^{-1}\)) | A1 |
| (6) | |
| (12 marks) |
Notes
First M1 for separating the variables and integrating
First A1 for a correct solution (without \(C\))
Second M1 for using conditions (or lower limits)
Second A1 for a correct particular solution in \(t\)
Third M1 for putting \(m = m_0\) and solving for \(t\).
Use of \(t = 2\) can imply this M mark
Third A1 for correct answer (allow \(2g\) instead of 19.6)