| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Work done and energy |
| Type | Resistance as function - finding acceleration or speed |
| Difficulty | Challenging +1.2 This is a standard M3 variable resistance problem requiring Newton's second law with power, followed by separating variables and integrating. Part (i) is routine algebraic manipulation to reach the given differential equation. Part (ii) requires integration of a rational function (likely partial fractions) and numerical solving, which is typical M3 fare but involves several careful steps. The setup is straightforward with no novel insight required, making it moderately above average difficulty for A-level but standard for M3. |
| Spec | 6.02l Power and velocity: P = Fv6.06a Variable force: dv/dt or v*dv/dx methods |
A motor-cycle, whose mass including the rider is $120$ kg, is decelerating on a horizontal straight road. The motor-cycle passes a point $A$ with speed $40 \text{ m s}^{-1}$ and when it has travelled a distance of $x$ m beyond $A$ its speed is $v \text{ m s}^{-1}$. The engine develops a constant power of $8$ kW and resistances are modelled by a force of $0.25v^2$ N opposing the motion.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1$. [5]
\item Find the speed of the motor-cycle when it has travelled $500$ m beyond $A$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR M3 2009 Q4 [11]}}