| Exam Board | OCR |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Air resistance kv - horizontal motion |
| Difficulty | Standard +0.8 This is a variable force mechanics problem requiring differential equations (separating variables for velocity-dependent force), followed by integration to find displacement. Part (i) is standard M3 material, but part (ii) requires finding displacement via v dv/dx = a, which demands careful technique selection and multi-step integration—above average difficulty but within expected M3 scope. |
| Spec | 3.02f Non-uniform acceleration: using differentiation and integration6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks |
|---|---|
| M1* | allow sign error |
| A1 | |
| *M1 | attempt to separate variables and integrate |
| A1 | \(x = 16\) when \(v = 5\). |
| *M1 | |
| A1 [6] | their \(x\) evaluated; accept 5/ln2 |
| M1* | for M1, fl from (i), must contain ln term |
| A1 | |
| *M1 | attempt to separate variables and integrate |
| A1 | must show constant or use limits correctly |
| *M1 | dep all 5 previous marks |
| A1 | accept 5/ln2 |
### Part (ii)
**Answer:** $\frac{v}{8} = 0.4v\frac{dv}{dx}$
$x = -3.2 \int dv$
$x = -3.2 v + 32$
ave speed = $x/(t)$
ave speed = 7.21
OR
$\frac{dx}{dt} = 10e^{-\frac{t}{3.2}}$
$x = 10 \int e^{-\frac{t}{3.2}} dt$
$x = 32\left(1 - e^{-\frac{t}{3.2}}\right)$
ave speed = $x/(t)$
ave speed = 7.21
| M1* | allow sign error |
| A1 | |
| *M1 | attempt to separate variables and integrate |
| A1 | $x = 16$ when $v = 5$. |
| *M1 | |
| A1 [6] | their $x$ evaluated; accept 5/ln2 |
| | |
| M1* | for M1, fl from (i), must contain ln term |
| A1 | |
| *M1 | attempt to separate variables and integrate |
| A1 | must show constant or use limits correctly |
| *M1 | dep all 5 previous marks |
| A1 | accept 5/ln2 |
**Guidance:** $x = 16$ when $t = 3.2\ln(2)$A particle of mass $0.4$ kg, moving on a smooth horizontal surface, passes through a point $O$ with velocity $10\text{ ms}^{-1}$. At time $t$ s after the particle passes through $O$, the particle has a displacement $x$ m from $O$, has a velocity $v\text{ ms}^{-1}$ away from $O$, and is acted on by a force of magnitude $\frac{1}{5}v$ N acting towards $O$. Find
\begin{enumerate}[label=(\roman*)]
\item the time taken for the velocity of the particle to reduce from $10\text{ ms}^{-1}$ to $5\text{ ms}^{-1}$, [5]
\item the average velocity of the particle over this time. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR M3 2015 Q4 [11]}}