| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable mass problems (mass increasing) |
| Difficulty | Challenging +1.8 This M4 question combines variable mass mechanics with differential equations involving air resistance. Part (i) requires solving a separable DE for mass loss. Part (ii) involves setting up and solving a non-standard equation of motion (F=ma with variable mass), requiring product rule manipulation and integrating factor technique to reach a specific form. Part (iii) is straightforward substitution. The variable mass aspect and the algebraic complexity of reaching the given answer elevate this significantly above typical mechanics questions, though it's structured with clear guidance. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.06a Variable force: dv/dt or v*dv/dx methods |
1 A raindrop of mass $m$ falls vertically from rest under gravity. Initially the mass of the raindrop is $m _ { 0 }$. As it falls it loses mass by evaporation at a rate $\lambda m$, where $\lambda$ is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is $v$ at time $t$. The forces on the raindrop are its weight and a resistance force of magnitude $k m v$, where $k$ is a constant.\\
(i) Find $m$ in terms of $m _ { 0 } , \lambda$ and $t$.\\
(ii) Write down the equation of motion of the raindrop. Solve this differential equation and hence show that $v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)$.\\
(iii) Find the velocity of the raindrop when it has lost half of its initial mass.
\hfill \mbox{\textit{OCR MEI M4 2011 Q1 [12]}}