| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Rocket ascending against gravity |
| Difficulty | Challenging +1.2 This is a standard variable mass/rocket equation problem from M5. Part (a) requires applying the rocket equation F = v_rel(dm/dt) + ma with gravity, which is a bookwork derivation. Part (b) involves separating variables and integrating with m = 1500 - 15t, requiring logarithmic integration. While it involves calculus and careful algebraic manipulation, it follows a well-established template for M5 rocket problems without requiring novel insight. |
| Spec | 4.10a General/particular solutions: of differential equations6.02m Variable force power: using scalar product6.06a Variable force: dv/dt or v*dv/dx methods |
A rocket, with initial mass 1500 kg, including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of 15 kg s$^{-1}$ and the burnt fuel is ejected vertically downwards with a speed of 1000 m s$^{-1}$ relative to the rocket. At time $t$ seconds after launch $(t \leqslant 40)$ the rocket has mass $m$ kg and velocity $v$ m s$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{dv}{dt} + \frac{1000}{m} \frac{dm}{dt} = -9.8$$
[5]
\end{enumerate}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find $v$ at time $t$, $0 \leqslant t \leqslant 40$
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q2 [10]}}