M5 June 2013 Q3
3. A raindrop falls vertically under gravity through a stationary cloud. At time \(t = 0\), the raindrop is at rest and has mass \(m_0\). As the raindrop falls, water condenses onto it from the cloud so that the mass of the raindrop increases at a constant rate \(c\). At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). The resistance to the motion of the raindrop has magnitude \(mkv\), where \(k\) is a constant. Show that
\[\frac{\mathrm{d}v}{\mathrm{d}t} + v\left(k + \frac{c}{m_0 + ct}\right) = g\](7)
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}m}{\mathrm{d}t} = c \Rightarrow m = m_0 + ct\) | |
| \((m + \delta m)(v + \delta v) - mv = (mg - mkv)\delta t\) | M1 A1 |
| \(m\delta v + v\delta m = (mg - mkv)\delta t\) | M1 A2 |
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} + kv + \dfrac{v}{m}\dfrac{\mathrm{d}m}{\mathrm{d}t} = g\) | |
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} + v\left(k + \dfrac{c}{m_0 + ct}\right) = g\) ** | DM1 A1 |
| (7 marks) |
Notes
First M1 for \(\dfrac{\mathrm{d}m}{\mathrm{d}t} = c\) and integrating
First A1 for \(m = m_0 + ct\)
Second M1 for impulse-momentum equation (correct number of terms, excluding any \(\delta m\delta v\) or \(\delta m\delta t\) terms)
Second and third A1: (-1 each error)
OR:
Second M1 for \(mg - mkv = \dfrac{\mathrm{d}}{\mathrm{d}t}(mv)\)
Second and third A1 for \(mg - mkv = v\dfrac{\mathrm{d}m}{\mathrm{d}t} + m\dfrac{\mathrm{d}v}{\mathrm{d}t}\) (-1 each error)
Third M1, dependent on second M1, for sub. for \(m\).
Third A1 for PRINTED ANSWER