| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Derive variable mass equation |
| Difficulty | Challenging +1.2 This is the classic rocket equation derivation (Tsiolkovsky equation), a standard M5 topic. Part (a) requires applying conservation of momentum to a variable mass system, which is conceptually non-trivial for A-level but follows a well-established method. Part (b) involves straightforward differentiation and substitution. While requiring careful reasoning about relative velocities and momentum, this is a textbook example that students preparing for M5 would practice extensively. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates6.03f Impulse-momentum: relation |
A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed $c$ relative to the spaceship. When the speed of the spaceship is $v$, its mass is $m$.
\begin{enumerate}[label=(\alph*)]
\item Show that, while the spaceship is ejecting fuel,
$$\frac{dv}{dm} = -\frac{c}{m}.$$
[5]
\end{enumerate}
The initial mass of the spaceship is $m_0$ and at time $t$ the mass of the spaceship is given by $m = m_0(1 - kt)$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the acceleration of the spaceship at time $t$.
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q3 [9]}}