| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2013 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Find unknown rate or constant |
| Difficulty | Challenging +1.2 This is a standard variable mass/rocket equation problem from M5. Part (a) requires applying the rocket equation derivation (a bookwork result that students practice), and part (b) involves straightforward integration and substitution. While the topic is advanced for A-level, this is a routine application rather than requiring novel insight, placing it moderately above average difficulty. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration6.03b Conservation of momentum: 1D two particles |
A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of 2000 m s$^{-1}$ relative to the spacecraft. The burnt fuel is ejected at a constant rate of $c$ kg s$^{-1}$. At time $t$ seconds the total mass of the spacecraft, including fuel, is $m$ kg and the speed of the spacecraft is $v$ m s$^{-1}$.
**(a)** Show that, while the spacecraft is ejecting burnt fuel,
$$m\frac{dv}{dt} = 2000c$$
(7 marks)
At time $t = 0$, the mass of the spacecraft is $M_0$ kg and the speed of the spacecraft is 2000 m s$^{-1}$. When $t = 50$, the spacecraft is still ejecting burnt fuel and its speed is 6000 m s$^{-1}$.
**(b)** Find $c$ in terms of $M_0$.
(7 marks)
---\begin{enumerate}
\item A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of $2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the spacecraft. The burnt fuel is ejected at a constant rate of $c \mathrm {~kg} \mathrm {~s} ^ { - 1 }$. At time $t$ seconds the total mass of the spacecraft, including fuel, is $m \mathrm {~kg}$ and the speed of the spacecraft is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(a) Show that, while the spacecraft is ejecting burnt fuel,
\end{enumerate}
$$m \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2000 c$$
At time $t = 0$, the mass of the spacecraft is $M _ { 0 } \mathrm {~kg}$ and the speed of the spacecraft is $2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. When $t = 50$, the spacecraft is still ejecting burnt fuel and its speed is $6000 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.\\
(b) Find $c$ in terms of $M _ { 0 }$.\\
\hfill \mbox{\textit{Edexcel M5 2013 Q3 [14]}}