| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Motion with exponential force |
| Difficulty | Standard +0.8 This is a variable force mechanics problem requiring Newton's second law with calculus (v dv/dx = a), integration of an exponential function, and numerical evaluation. While the setup is standard M2 fare, the exponential force and multi-step integration make it moderately above average difficulty for A-level, though still within typical Further Maths M2 scope. |
| Spec | 4.10a General/particular solutions: of differential equations6.06a Variable force: dv/dt or v*dv/dx methods |
| Answer | Marks |
|---|---|
| (ii) | 0.4vdv/dx = 0.4 g sin30 – 0.8e−x |
| Answer | Marks |
|---|---|
| v = 2.05 | M1 |
| Answer | Marks |
|---|---|
| A1 | 2 |
| 4 | Separates the variables and attempts to |
Question 3:
--- 3 (i)
(ii) ---
3 (i)
(ii) | 0.4vdv/dx = 0.4 g sin30 – 0.8e−x
vdv/dx = 5 – 2e−x AG
∫vdv = ∫(5−2e−x ) dx
v2 /2 = 5x + 2e−x ( + c )
v = 2.05 | M1
A1
M1
A1
M1
A1 | 2
4 | Separates the variables and attempts to
integrate
Uses limits or finds c (c = –2)A particle $P$ of mass $0.4 \text{ kg}$ is released from rest at a point $O$ on a smooth plane inclined at $30°$ to the horizontal. When the displacement of $P$ from $O$ is $x \text{ m}$ down the plane, the velocity of $P$ is $v \text{ ms}^{-1}$. A force of magnitude $0.8e^{-x} \text{ N}$ acts on $P$ up the plane along the line of greatest slope through $O$.
\begin{enumerate}[label=(\roman*)]
\item Show that $v \frac{dv}{dx} = 5 - 2e^{-x}$. [2]
\item Find $v$ when $x = 0.6$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2016 Q3 [6]}}