| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2014 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Derive variable mass equation |
| Difficulty | Challenging +1.8 This is a challenging M5 variable mass/rocket propulsion question requiring derivation of the rocket equation from first principles, integration to find mass at rest, and a multi-step distance calculation involving exponential mass variation. While the techniques are standard for M5, the question demands careful application of conservation of momentum, separation of variables, and substitution across multiple parts with significant algebraic manipulation. |
| Spec | 4.10a General/particular solutions: of differential equations4.10b Model with differential equations: kinematics and other contexts6.06a Variable force: dv/dt or v*dv/dx methods |
A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed $k$ relative to the spacecraft, in the direction of motion of the spacecraft. At time $t$, the spacecraft has speed $v$ and mass $m$.
\begin{enumerate}[label=(\alph*)]
\item Show, from first principles, that while the spacecraft is ejecting fuel,
$$\frac{dv}{dm} - \frac{k}{m} = 0$$
[5]
\end{enumerate}
At time $t = 0$, the spacecraft has speed $U$ and mass $M$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the mass of the spacecraft when it comes to rest.
[6]
\end{enumerate}
Given that $m = Me^{-\alpha t^2}$, where $\alpha$ is a positive constant, and that the spacecraft comes to rest at time $t = T$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find, in terms of $U$ and $T$ only, the distance travelled by the spacecraft in decelerating from speed $U$ to rest.
[6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2014 Q4 [17]}}