| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable mass problems (mass increasing) |
| Difficulty | Challenging +1.8 This is a challenging M4 variable mass problem requiring students to set up differential equations from physical principles (mass gain proportional to surface area), manipulate the rocket equation for a gaining-mass system, and integrate carefully. It demands strong conceptual understanding beyond standard mechanics and involves non-trivial multi-step reasoning with unfamiliar physical contexts. |
| Spec | 4.10a General/particular solutions: of differential equations6.06a Variable force: dv/dt or v*dv/dx methods |
1 A spherical raindrop falls through a stationary cloud. Water condenses on the raindrop and it gains mass at a rate proportional to its surface area. At time $t$ the radius of the raindrop is $r$. Initially the raindrop is at rest and $r = r _ { 0 }$. The density of the water is $\rho$.\\
(i) Show that $\frac { \mathrm { d } r } { \mathrm {~d} t } = k$, where $k$ is a constant. Hence find the mass of the raindrop in terms of $r _ { 0 } , \rho , k$ and $t$.\\
(ii) Assuming that air resistance is negligible, find the velocity of the raindrop in terms of $r _ { 0 } , k$ and $t$.
\hfill \mbox{\textit{OCR MEI M4 2006 Q1 [12]}}