| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2006 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable mass problems |
| Type | Find ejected or remaining mass |
| Difficulty | Challenging +1.8 This is a variable mass mechanics problem requiring derivation of the rocket equation from first principles using momentum conservation, followed by integration and interpretation. M5 is an advanced Further Maths module, and while the rocket equation is a standard result, deriving it requires careful handling of relative velocities and differential equations. The 6+6 mark allocation indicates substantial working, but the problem follows a well-established framework without requiring novel geometric or algebraic insight beyond the standard approach. |
| Spec | 4.10a General/particular solutions: of differential equations6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks |
|---|---|
| \(\frac{dm}{dv} = \frac{m}{k}\) * | M1 A3 M1 A1(6) |
| Answer | Marks |
|---|---|
| Amount of fuel \(= M - m_1 = M\left(1 - e^{\frac{-v}{k}}\right)\) | M1 A1 M1 A1 M1 A1(6)(12) |
## Part (a)
$m\ddot{r} = (m+\delta m)(\dot{v}+\delta\dot{v}) + (-\delta m)(v+\dot{v}+\delta\dot{v})$
$m\dot{v} = m\dot{v} + m\delta\dot{v} + v\delta m - v\delta m - \dot{v}\delta m$
$k\delta m = m\delta\dot{v}$
In the limit, as $\delta t \to 0$,
$\frac{dm}{dv} = \frac{m}{k}$ * | M1 A3 M1 A1(6) |
## Part (b)
$u_1 \int \frac{dm}{m} = \int \frac{dv}{k}$
$\ln u_1 - \ln M = \frac{1}{k}(v-u)$
$\ln \frac{m}{M} = \frac{1}{k}(v-u)$
$m_1 = Me^{\frac{v-u}{k}}$
Amount of fuel $= M - m_1 = M\left(1 - e^{\frac{-v}{k}}\right)$ | M1 A1 M1 A1 M1 A1(6)(12) |
---A space-ship is moving in a straight line in deep space and needs to reduce its speed from $U$ to $V$. This is done by ejecting fuel from the front of the space-ship at a constant speed $k$ relative to the space-ship. When the speed of the space-ship is $v$, its mass is $m$.
\begin{enumerate}[label=(\alph*)]
\item Show that, while the space-ship is ejecting fuel, $\frac{\mathrm{d}m}{\mathrm{d}v} = -\frac{m}{k}$. [6]
\end{enumerate}
The initial mass of the space-ship is $M$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, in terms of $U$, $V$, $k$ and $M$, the amount of fuel which needs to be used to reduce the speed of the space-ship from $U$ to $V$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2006 Q5 [12]}}