| Exam Board | AQA |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable Force |
| Type | Force depends on time t |
| Difficulty | Standard +0.3 This is a standard M2 variable force question requiring Newton's second law, integration of trigonometric and polynomial functions, and vector manipulation. While it has multiple parts, each step follows routine procedures: F=ma for acceleration, integration with boundary conditions for velocity, solving when j-component dominates for direction, and finding magnitude. The integration is straightforward (cos and t² terms) with no conceptual surprises, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits3.02f Non-uniform acceleration: using differentiation and integration3.03d Newton's second law: 2D vectors |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{1}{200}\left(400\cos\left(\frac{\pi}{2}t\right)\mathbf{i} + 600t^2\mathbf{j}\right)\) | M1 | Use of \(F = ma\) |
| \(\mathbf{a} = \left(2\cos\left(\frac{\pi}{2}t\right)\mathbf{i} + 3t^2\mathbf{j}\right) \text{ m s}^{-2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{v} = \int \mathbf{a} \, dt = \frac{2\sin\left(\frac{\pi}{2}t\right)}{\frac{\pi}{2}}\mathbf{i} + t^3\mathbf{j} + \mathbf{c}\) | M1 | Integrate acceleration |
| \(= \frac{4}{\pi}\sin\left(\frac{\pi}{2}t\right)\mathbf{i} + t^3\mathbf{j} + \mathbf{c}\) | A1 | Correct integration |
| When \(t=4\): \(-3\mathbf{i} + 56\mathbf{j} = \frac{4}{\pi}\sin(2\pi)\mathbf{i} + 64\mathbf{j} + \mathbf{c}\) | M1 | Substitute \(t=4\) |
| \(\mathbf{c} = -3\mathbf{i} - 8\mathbf{j}\) | A1 | |
| \(\mathbf{v} = \left(\frac{4}{\pi}\sin\left(\frac{\pi}{2}t\right) - 3\right)\mathbf{i} + \left(t^3 - 8\right)\mathbf{j}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Moving due west: \(\mathbf{j}\) component \(= 0\) and \(\mathbf{i}\) component \(< 0\) | M1 | |
| \(t^3 - 8 = 0 \Rightarrow t = 2\) | A1 | |
| Check \(\mathbf{i}\) component: \(\frac{4}{\pi}\sin(\pi) - 3 = -3 < 0\) ✓ | A1 | Confirm moving west |
| Answer | Marks | Guidance |
|---|---|---|
| Speed \(= \left | \frac{4}{\pi}\sin(\pi) - 3\right | = |
| \(= 3 \text{ m s}^{-1}\) | A1 |
# Question 4:
## Part (a)
| $\mathbf{a} = \frac{\mathbf{F}}{m} = \frac{1}{200}\left(400\cos\left(\frac{\pi}{2}t\right)\mathbf{i} + 600t^2\mathbf{j}\right)$ | M1 | Use of $F = ma$ |
|---|---|---|
| $\mathbf{a} = \left(2\cos\left(\frac{\pi}{2}t\right)\mathbf{i} + 3t^2\mathbf{j}\right) \text{ m s}^{-2}$ | A1 | |
## Part (b)
| $\mathbf{v} = \int \mathbf{a} \, dt = \frac{2\sin\left(\frac{\pi}{2}t\right)}{\frac{\pi}{2}}\mathbf{i} + t^3\mathbf{j} + \mathbf{c}$ | M1 | Integrate acceleration |
|---|---|---|
| $= \frac{4}{\pi}\sin\left(\frac{\pi}{2}t\right)\mathbf{i} + t^3\mathbf{j} + \mathbf{c}$ | A1 | Correct integration |
| When $t=4$: $-3\mathbf{i} + 56\mathbf{j} = \frac{4}{\pi}\sin(2\pi)\mathbf{i} + 64\mathbf{j} + \mathbf{c}$ | M1 | Substitute $t=4$ |
| $\mathbf{c} = -3\mathbf{i} - 8\mathbf{j}$ | A1 | |
| $\mathbf{v} = \left(\frac{4}{\pi}\sin\left(\frac{\pi}{2}t\right) - 3\right)\mathbf{i} + \left(t^3 - 8\right)\mathbf{j}$ | A1 | |
## Part (c)
| Moving due west: $\mathbf{j}$ component $= 0$ and $\mathbf{i}$ component $< 0$ | M1 | |
|---|---|---|
| $t^3 - 8 = 0 \Rightarrow t = 2$ | A1 | |
| Check $\mathbf{i}$ component: $\frac{4}{\pi}\sin(\pi) - 3 = -3 < 0$ ✓ | A1 | Confirm moving west |
## Part (d)
| Speed $= \left|\frac{4}{\pi}\sin(\pi) - 3\right| = |-3| = 3 \text{ m s}^{-1}$ | M1 | Substitute their $t$ |
|---|---|---|
| $= 3 \text{ m s}^{-1}$ | A1 | |
---4 A particle has mass 200 kg and moves on a smooth horizontal plane. A single horizontal force, $\left( 400 \cos \left( \frac { \pi } { 2 } t \right) \mathbf { i } + 600 t ^ { 2 } \mathbf { j } \right)$ newtons, acts on the particle at time $t$ seconds.
The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the acceleration of the particle at time $t$.
\item When $t = 4$, the velocity of the particle is $( - 3 \mathbf { i } + 56 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$.
Find the velocity of the particle at time $t$.
\item Find $t$ when the particle is moving due west.
\item Find the speed of the particle when it is moving due west.\\
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-09_2484_1709_223_153}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{AQA M2 2010 Q4 [12]}}