| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Variable force (position x) - find velocity |
| Difficulty | Standard +0.3 This is a standard variable force mechanics problem requiring Newton's second law with F=ma=m(v dv/dx), followed by straightforward integration and substitution. The exponential function adds minor complexity, but the steps are routine for M2 level with clear signposting and no novel problem-solving required. |
| Spec | 4.10a General/particular solutions: of differential equations6.06a Variable force: dv/dt or v*dv/dx methods |
A particle $P$ of mass $0.1$ kg moves with decreasing speed in a straight line on a smooth horizontal surface. A horizontal resisting force of magnitude $0.2e^{-x}$ N acts on $P$, where $x$ m is the displacement of $P$ from a fixed point $O$ on the line. The velocity of $P$ is $v$ m s$^{-1}$ when its displacement from $O$ is $x$ m.
\begin{enumerate}[label=(\roman*)]
\item Show that
$$v\frac{dv}{dx} = ke^{-x},$$
where $k$ is a constant to be found. [2]
\end{enumerate}
$P$ passes through $O$ with velocity $2.2$ m s$^{-1}$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Calculate the value of $x$ at the instant when the velocity of $P$ is $2$ m s$^{-1}$. [4]
\item Show that the speed of $P$ does not fall below $0.917$ m s$^{-1}$, correct to $3$ significant figures. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2015 Q6 [8]}}