| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Displacement from velocity by integration |
| Difficulty | Moderate -0.8 This is a straightforward mechanics question requiring standard differentiation and integration of given functions. Part (a) involves differentiating a polynomial and trigonometric function, then substituting a value. Part (b) requires integrating the velocity function with a boundary condition. All techniques are routine M2 content with no problem-solving insight needed. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(a = \frac{dv}{dt} = 6t - 6\cos 3t\) | M1A1 | 2 marks |
| (ii) When \(t = \frac{\pi}{3}\), \(a = 6 \times \frac{\pi}{3} - 6\cos(3 \cdot \frac{\pi}{3}) = 2\pi + 6\) | M1 A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\therefore r = t^3 + \frac{2}{3}\cos 3t + 6t - \frac{2}{3}\) | M1A1 M1 A1 | 4 marks |
**(a)(i)** $a = \frac{dv}{dt} = 6t - 6\cos 3t$ | M1A1 | 2 marks | M1 for at least one term correct
**(ii)** When $t = \frac{\pi}{3}$, $a = 6 \times \frac{\pi}{3} - 6\cos(3 \cdot \frac{\pi}{3}) = 2\pi + 6$ | M1 A1 | 2 marks | AG
**(b)** $r = t^3 + \frac{2}{3}\cos 3t + 6t + c$
When $t = 0$, $r = 0 \therefore c = -\frac{2}{3}$
$\therefore r = t^3 + \frac{2}{3}\cos 3t + 6t - \frac{2}{3}$ | M1A1 M1 A1 | 4 marks | M1 for 3 terms including $\cos 3t$ term; Condone no '+ c'
**Total: 8 marks**
---2 A particle moves in a straight line and at time $t$ it has velocity $v$, where
$$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find an expression for the acceleration of the particle at time $t$.
\item When $t = \frac { \pi } { 3 }$, show that the acceleration of the particle is $2 \pi + 6$.
\end{enumerate}\item When $t = 0$, the particle is at the origin.
Find an expression for the displacement of the particle from the origin at time $t$.
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2008 Q2 [8]}}