| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2024 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Angle change from impulse |
| Difficulty | Moderate -0.8 This is a straightforward impulse-momentum question requiring standard vector manipulation: apply impulse equation to find new velocity, calculate speed using Pythagoras, then find angle change using dot product or arctangent. All steps are routine M2 techniques with no problem-solving insight needed, making it easier than average but not trivial due to the two-part vector calculation. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication6.03f Impulse-momentum: relation6.03g Impulse in 2D: vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use of \(\mathbf{I} = m\mathbf{v} - m\mathbf{u}\) | M1 | Column vectors acceptable. Condone wrong order but must be subtracting. Condone 5 in place of 0.5 |
| \(2\mathbf{i} + 5\mathbf{j} = 0.5\left(\mathbf{v} - (3\mathbf{i} + \mathbf{j})\right)\) \(\left(\mathbf{v} = 7\mathbf{i} + 11\mathbf{j}\right)\) | A1 | Correct unsimplified equation. Accept as a vector equation or as a pair of equations |
| Use of Pythagoras | M1 | For their \(\mathbf{v}\). Independent M1 but they must have a \(\mathbf{v}\) |
| \(\ | v\ | = \sqrt{121 + 49} = \sqrt{170}\ \left(\mathrm{ms}^{-1}\right)\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct use of trigonometry e.g. \(\theta = \tan^{-1}\frac{11}{7} - \tan^{-1}\frac{1}{3}\) \((= 57.5 - 18.4)\) | M1 | Condone subtraction in either order. Allow if both fractions are the other way up. Alternatives: scalar product \(\theta = \cos^{-1}\left(\frac{21+11}{\sqrt{10}\sqrt{170}}\right)\), cosine rule \(4\times29 = 10+170-2\sqrt{10}\sqrt{170}\cos\theta\) |
| \(\theta = 39.1\) | A1 | Accept \(\pm 39\) or better \((39.0938\ldots)\). \(0.68(2)\) radians is M1A0. Accept \(\pm(360-39) = \pm321\) or better |
| [2] |
## Question 2:
**Part 2a:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use of $\mathbf{I} = m\mathbf{v} - m\mathbf{u}$ | M1 | Column vectors acceptable. Condone wrong order but must be subtracting. Condone 5 in place of 0.5 |
| $2\mathbf{i} + 5\mathbf{j} = 0.5\left(\mathbf{v} - (3\mathbf{i} + \mathbf{j})\right)$ $\left(\mathbf{v} = 7\mathbf{i} + 11\mathbf{j}\right)$ | A1 | Correct unsimplified equation. Accept as a vector equation or as a pair of equations |
| Use of Pythagoras | M1 | For their $\mathbf{v}$. Independent M1 but they must have a $\mathbf{v}$ |
| $\|v\| = \sqrt{121 + 49} = \sqrt{170}\ \left(\mathrm{ms}^{-1}\right)$ | A1 | $13\left(\mathrm{ms}^{-1}\right)$ or better. $(13.038\ldots)$ |
| **[4]** | | |
**Part 2b:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct use of trigonometry e.g. $\theta = \tan^{-1}\frac{11}{7} - \tan^{-1}\frac{1}{3}$ $(= 57.5 - 18.4)$ | M1 | Condone subtraction in either order. Allow if **both** fractions are the other way up. Alternatives: scalar product $\theta = \cos^{-1}\left(\frac{21+11}{\sqrt{10}\sqrt{170}}\right)$, cosine rule $4\times29 = 10+170-2\sqrt{10}\sqrt{170}\cos\theta$ |
| $\theta = 39.1$ | A1 | Accept $\pm 39$ or better $(39.0938\ldots)$. $0.68(2)$ radians is M1A0. Accept $\pm(360-39) = \pm321$ or better |
| **[2]** | | |
---\begin{enumerate}
\item \hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal perpendicular unit vectors.]
\end{enumerate}
A particle $Q$ of mass 0.5 kg is moving on a smooth horizontal surface. Particle $Q$ is moving with velocity $( 3 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ when it receives an impulse of $( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { Ns }$.\\
(a) Find the speed of $Q$ immediately after receiving the impulse.
As a result of receiving the impulse, the direction of motion of $Q$ is turned through an angle $\theta ^ { \circ }$\\
(b) Find the value of $\theta$
\hfill \mbox{\textit{Edexcel M2 2024 Q2 [6]}}