| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2018 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Variable force with integration (force as function of position) |
| Difficulty | Challenging +1.2 This is a multi-step mechanics problem requiring energy methods and differential equations. Part (i) involves applying Newton's second law with variable forces (elastic tension and position-dependent resistance), which is standard M2 content. Part (ii) requires integrating the differential equation and applying boundary conditions. While it involves several techniques and careful algebraic manipulation, the approach is methodical and follows standard M2 procedures without requiring novel insight. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.06a Variable force: dv/dt or v*dv/dx methods |
\includegraphics{figure_4}
A particle $P$ of mass $0.5\text{ kg}$ is projected along a smooth horizontal surface towards a fixed point $A$. Initially $P$ is at a point $O$ on the surface, and after projection, $P$ has a displacement from $O$ of $x\text{ m}$ and velocity $v\text{ m s}^{-1}$. The particle $P$ is connected to $A$ by a light elastic string of natural length $0.8\text{ m}$ and modulus of elasticity $16\text{ N}$. The distance $OA$ is $1.6\text{ m}$ (see diagram). The motion of $P$ is resisted by a force of magnitude $24x^2\text{ N}$.
\begin{enumerate}[label=(\roman*)]
\item Show that $v\frac{\text{d}v}{\text{d}x} = 32 - 40x - 48x^2$ while $P$ is in motion and the string is stretched. [3]
The maximum value of $v$ is $4.5$.
\item Find the initial value of $v$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2018 Q4 [8]}}