| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Acceleration as function of velocity (separation of variables) |
| Difficulty | Standard +0.3 This is a standard mechanics question involving Newton's second law with resistance proportional to velocity. Part (i) is straightforward force equation manipulation (2 marks). Part (ii) requires separating variables and integrating a simple linear differential equation, which is a routine M2 technique, though the integration and algebra require care. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.2 \, dv/dt = 0.2g - 0.8v\) | M1 | Use Newton's Second Law, – sign essential |
| \(a = (dv/dt) = 10 - 4v\) | AG A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int 1/(10 - 4v) \, dv = \int dt\) | M1 | Separates variables and attempts to integrate |
| \(-\frac{1}{4} \ln(10 - 4v) = t (+c)\) | A1 | |
| \([c = -\frac{1}{4} \ln 10]\) | M1 | Attempts to find the constant or uses the correct limits |
| \(-\frac{1}{4} \ln(10 - 4v) = 0.6 - \frac{1}{4} \ln4\) | A1 | |
| \(v = 2.27\) | A1 [5] |
## (i)
$0.2 \, dv/dt = 0.2g - 0.8v$ | M1 | Use Newton's Second Law, – sign essential
$a = (dv/dt) = 10 - 4v$ | AG A1 [2] |
## (ii)
$\int 1/(10 - 4v) \, dv = \int dt$ | M1 | Separates variables and attempts to integrate
$-\frac{1}{4} \ln(10 - 4v) = t (+c)$ | A1 |
$[c = -\frac{1}{4} \ln 10]$ | M1 | Attempts to find the constant or uses the correct limits
$-\frac{1}{4} \ln(10 - 4v) = 0.6 - \frac{1}{4} \ln4$ | A1 |
$v = 2.27$ | A1 [5] |A particle $P$ of mass $0.2$ kg is released from rest and falls vertically. At time $t$ s after release $P$ has speed $v$ m s$^{-1}$. A resisting force of magnitude $0.8v$ N acts on $P$.
\begin{enumerate}[label=(\roman*)]
\item Show that the acceleration of $P$ is $(10 - 4v)$ m s$^{-2}$. [2]
\item Find the value of $v$ when $t = 0.6$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2012 Q3 [7]}}