| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on time t |
| Difficulty | Standard +0.3 This is a straightforward variable force problem requiring integration of F=ma with a simple linear force function. Students apply Newton's second law to get dv/dt = -12t, integrate once to find initial velocity using the boundary condition, then integrate again to find distance. All steps are standard M2 techniques with no conceptual surprises. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.25v\frac{dv}{dt} = -3t\) | M1 | Newton's Second Law, – sign essential |
| \(v = -12t^2/(2 + c)\) | A1 | Accept uncancelled form |
| \(0 = 12 \times 3^2/2 + c\) | M1 | Appropriate use of \(v = 0\), \(t = 3\) |
| Initial speed \(= 54\,\mathrm{ms}^{-1}\) | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int dx = \int [54 - 6t^2]\,dt\) | M1 | Separates variables, integrates \(v\) |
| \(x = [54t - 6t^3/3]_0^3\) | A1 | \(\unicode{x2713}\) candidates value [\(v\) in (i)] |
| \(x = 108\,\mathrm{m}\) | A1 | [3] |
**(i)**
$0.25v\frac{dv}{dt} = -3t$ | M1 | Newton's Second Law, – sign essential
$v = -12t^2/(2 + c)$ | A1 | Accept uncancelled form
$0 = 12 \times 3^2/2 + c$ | M1 | Appropriate use of $v = 0$, $t = 3$
Initial speed $= 54\,\mathrm{ms}^{-1}$ | A1 | [4] | Goes beyond $c = 54$
**(ii)**
$\int dx = \int [54 - 6t^2]\,dt$ | M1 | Separates variables, integrates $v$
$x = [54t - 6t^3/3]_0^3$ | A1 | $\unicode{x2713}$ candidates value [$v$ in (i)]
$x = 108\,\mathrm{m}$ | A1 | [3] | [7]4 A particle $P$ of mass 0.25 kg moves in a straight line on a smooth horizontal surface. At time $t \mathrm {~s}$ the velocity of $P$ is $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A variable force of magnitude $3 t \mathrm {~N}$ opposes the motion of $P$.\\
(i) Given that $P$ comes to rest when $t = 3$, find $v$ when $t = 0$.\\
(ii) Calculate the distance travelled by $P$ in the interval $0 \leqslant t \leqslant 3$.
\hfill \mbox{\textit{CAIE M2 2012 Q4 [7]}}