| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable mass problems (mass increasing) |
| Difficulty | Challenging +1.8 This is a challenging M5 variable mass problem requiring application of Newton's second law in the form F = d(mv)/dt, careful handling of the product rule with time-varying mass m = m₀ + ct, and algebraic manipulation to reach the given differential equation. It demands sophisticated mechanics understanding beyond standard A-level, though the derivation itself is relatively structured once the correct approach is identified. |
| Spec | 6.03b Conservation of momentum: 1D two particles |
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
$$\frac{dv}{dt} + \left(\frac{c}{m_0 + ct} + k\right)v = g$$
(7 marks)
---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 $m k v$, where $k$ is a constant. Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } + v \left( k + \frac { c } { m _ { 0 } + c t } \right) = g$$
\hfill \mbox{\textit{Edexcel M5 2013 Q3 [7]}}