| 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 | Velocity from acceleration by integration |
| Difficulty | Standard +0.8 This is a standard M2 non-constant acceleration problem requiring Newton's second law with resistance proportional to velocity, followed by separating variables and integrating. Part (i) is straightforward force balance; part (ii) requires setting up and solving dv/dt = 10-4v, which is a routine technique for this module. More challenging than basic mechanics but standard for M2 level. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.2 \frac{dv}{dt} = 0.2g - 0.8v\) | M1 | Use Newton's Second Law, — sign essential |
| \(a = (dv/dt) = 10 - 4v\) | AG A1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{1}{(10-4v)} dv = \int dt\) | 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)**
$0.2 \frac{dv}{dt} = 0.2g - 0.8v$ | M1 | Use Newton's Second Law, — sign essential
$a = (dv/dt) = 10 - 4v$ | AG A1 [2] |
**(ii)**
$\int \frac{1}{(10-4v)} dv = \int dt$ | 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] |3 A particle $P$ of mass 0.2 kg is released from rest and falls vertically. At time $t \mathrm {~s}$ after release $P$ has speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A resisting force of magnitude $0.8 v \mathrm {~N}$ acts on $P$.\\
(i) Show that the acceleration of $P$ is $( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }$.\\
(ii) Find the value of $v$ when $t = 0.6$.
\hfill \mbox{\textit{CAIE M2 2012 Q3 [7]}}