| Exam Board | OCR MEI |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Rocket in deep space, no gravity |
| Difficulty | Challenging +1.2 This is a standard variable mass rocket equation derivation from M4/Further Mechanics. Part (i) requires applying the rocket equation (momentum conservation for variable mass systems) - a bookwork result that students are expected to know. Part (ii) is a straightforward separable differential equation. While the topic is advanced (Further Maths), the execution is mechanical and follows a well-rehearsed procedure 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 A rocket in deep space has initial mass $m _ { 0 }$ and is moving in a straight line at speed $v _ { 0 }$. It fires its engine in the direction opposite to the motion in order to increase its speed. The propulsion system ejects matter at a constant mass rate $k$ with constant speed $u$ relative to the rocket. At time $t$ after the engines are fired, the speed of the rocket is $v$.\\
(i) Show that while mass is being ejected from the rocket, $\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k$.\\
(ii) Hence find an expression for $v$ at time $t$.
\hfill \mbox{\textit{OCR MEI M4 2012 Q1 [11]}}