| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Rocket ascending against gravity |
| Difficulty | Challenging +1.2 This is a standard variable mass rocket equation derivation followed by integration with given initial conditions. Part (i) requires applying Newton's second law to a variable mass system (a key M4 technique but well-rehearsed), and part (ii) involves straightforward integration and substitution. While conceptually more sophisticated than basic mechanics, this is a textbook application of the rocket equation with no novel problem-solving required. |
| Spec | 6.03b Conservation of momentum: 1D two particles6.06a Variable force: dv/dt or v*dv/dx methods |
1 At time $t$ a rocket has mass $m$ and is moving vertically upwards with velocity $v$. The propulsion system ejects matter at a constant speed $u$ relative to the rocket. The only additional force acting on the rocket is its weight.\\
(i) Derive the differential equation $m \frac { \mathrm {~d} v } { \mathrm {~d} t } + u \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g$.
The rocket has initial mass $m _ { 0 }$ of which $75 \%$ is fuel. It is launched from rest. Matter is ejected at a constant mass rate $k$.\\
(ii) Assuming that the acceleration due to gravity is constant, find the speed of the rocket at the instant when all the fuel is burnt.
\hfill \mbox{\textit{OCR MEI M4 2010 Q1 [12]}}