| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Finding when moving in specific direction |
| Difficulty | Moderate -0.3 This is a straightforward M2 question testing standard techniques: differentiation of position vectors for velocity (routine), solving when j-component equals zero (algebraic manipulation), and applying impulse-momentum theorem (direct formula application). All parts are textbook exercises requiring no problem-solving insight, though the multi-step nature and impulse calculation elevate it slightly above pure recall. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors3.02a Kinematics language: position, displacement, velocity, acceleration6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dot{\mathbf{p}} = (6t - 6)\mathbf{i} + (9t^2 - 4)\mathbf{j}\) (ms\(^{-1}\)) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(9t^2 - 4 = 0\), \(t = \frac{2}{3}\) | M1 DM1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{v} = 4\mathbf{i} - 7\mathbf{j}\) (ms\(^{-1}\)) | B1 ft M1 M1 A1 | (4 marks) [9 marks total] |
**Part (a)**
$\dot{\mathbf{p}} = (6t - 6)\mathbf{i} + (9t^2 - 4)\mathbf{j}$ (ms$^{-1}$) | M1 A1 | (2 marks)
**Part (b)**
$9t^2 - 4 = 0$, $t = \frac{2}{3}$ | M1 DM1 A1 | (3 marks)
**Part (c)**
$t = 1 \Rightarrow \dot{\mathbf{p}} = 5\mathbf{j}$
(+/-) $2\mathbf{i} - 6\mathbf{j} = 0.5(\mathbf{v} - 5\mathbf{j})$
$\mathbf{v} = 4\mathbf{i} - 7\mathbf{j}$ (ms$^{-1}$) | B1 ft M1 M1 A1 | (4 marks) [9 marks total]
Fits their $\dot{\mathbf{p}}$
---At time $t$ seconds $(t \geq 0)$, a particle $P$ has position vector $\mathbf{p}$ metres, with respect to a fixed origin $O$, where
$$\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.$$
Find
\begin{enumerate}[label=(\alph*)]
\item the velocity of $P$ at time $t$ seconds, [2]
\item the value of $t$ when $P$ is moving parallel to the vector $\mathbf{i}$. [3]
\end{enumerate}
When $t = 1$, the particle $P$ receives an impulse of $(2\mathbf{i} - 6\mathbf{j})$ N s. Given that the mass of $P$ is 0.5 kg,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the velocity of $P$ immediately after the impulse. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2008 Q2 [9]}}