| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Power and driving force |
| Type | Variable resistance: find constant speed |
| Difficulty | Standard +0.8 Part (a) is a standard derivation using F=ma and power=force×velocity (3 marks of routine setup). Part (b) requires separating variables in a non-trivial differential equation, partial fractions decomposition, integration, and algebraic manipulation to reach a final answer—this is a substantial M4 problem requiring multiple techniques and careful algebra, placing it moderately above average difficulty. |
| Spec | 4.10b Model with differential equations: kinematics and other contexts6.02l Power and velocity: P = Fv |
A train of mass $m$ is moving along a straight horizontal railway line. A time $t$, the train is moving with speed $v$ and the resistance to motion has magnitude $kv$, where $k$ is a constant. The engine of the train is working at a constant rate $P$.
\begin{enumerate}[label=(\alph*)]
\item Show that, when $v > 0$, $mv\frac{dv}{dt} + kv^2 = P$.
[3]
\end{enumerate}
When $t = 0$, the speed of the train is $\frac{1}{3}\sqrt{\frac{P}{k}}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, in terms of $m$ and $k$, the time taken for the train to double its initial speed.
[8]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2006 Q5 [11]}}