| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| Paper | 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 velocity to find acceleration, application of F=ma, and integration with boundary conditions. All techniques are routine calculus and vector operations with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form3.02a Kinematics language: position, displacement, velocity, acceleration3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors |
2 A particle moves in a horizontal plane. The vectors $\mathbf { i }$ and $\mathbf { j }$ are perpendicular unit vectors in the horizontal plane. At time $t$ seconds, the velocity of the particle, $\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }$, is given by
$$\mathbf { v } = 12 \cos \left( \frac { \pi } { 3 } t \right) \mathbf { i } - 9 t ^ { 2 } \mathbf { j }$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for the acceleration of the particle at time $t$.
\item The particle, which has mass 4 kg , moves under the action of a single force, $\mathbf { F }$ newtons.
\begin{enumerate}[label=(\roman*)]
\item Find an expression for the force $\mathbf { F }$ in terms of $t$.
\item Find the magnitude of $\mathbf { F }$ when $t = 3$.
\end{enumerate}\item When $t = 3$, the particle is at the point with position vector $( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }$.
Find the position vector, $\mathbf { r }$ metres, of the particle at time $t$.
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2013 Q2 [11]}}