| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (vectors) |
| Type | Find force using F=ma |
| Difficulty | Standard +0.3 This is a straightforward M2 mechanics question requiring standard differentiation of position vectors twice to find force, then basic vector magnitude and direction calculations. All steps are routine applications of F=ma with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(v = (3t^2 - 6t)i + (4 + 2t)j\) | M1A1 | 2 marks |
| (b)(i) \(a = (6t - 6)i + 2j\) | M1 A1ft | 3 marks |
| Answer | Marks |
|---|---|
| \(F = (18t - 18)i + 6j\) | A1ft |
| Answer | Marks | Guidance |
|---|---|---|
| Magnitude is \(\sqrt{36^2 + 6^2} = 36.5\) | M1 A1ft | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = 1\) | M1 A1ft | 2 marks |
**(a)** $v = \frac{dr}{dt}$
$v = (3t^2 - 6t)i + (4 + 2t)j$ | M1A1 | 2 marks
**(b)(i)** $a = (6t - 6)i + 2j$ | M1 A1ft | 3 marks
Using F = ma:
$F = (18t - 18)i + 6j$ | A1ft
**(ii)** When $t = 3$, $F = 36i + 6j$
Magnitude is $\sqrt{36^2 + 6^2} = 36.5$ | M1 A1ft | 2 marks | Accept $6\sqrt{37}$; ft from (b)(i)
**(c)** When F acts due north:
Component of F in the i direction is 0
$18t - 18 = 0$
$t = 1$ | M1 A1ft | 2 marks | ft from (b)(i)
**Total: 9 marks**
---4 A particle moves in a horizontal plane under the action of a single force, $\mathbf { F }$ newtons. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively. At time $t$ seconds, the position vector, $\mathbf { r }$ metres, of the particle is given by
$$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for the velocity of the particle at time $t$.
\item The mass of the particle is 3 kg .
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\mathbf { F }$ at time $t$.
\item Find the magnitude of $\mathbf { F }$ when $t = 3$.
\end{enumerate}\item Find the value of $t$ when $\mathbf { F }$ acts due north.
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2008 Q4 [9]}}