A2 June 2022 Q3
3.
A particle \(P\) of mass 0.5 kg is moving in a straight line with speed \(2.8\ \text{m s}^{-1}\) when it receives an impulse of magnitude 3 N s.
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the line of action of the impulse is \(\alpha\), where \(\tan\alpha = \dfrac{4}{3}\), as shown in Figure 2.
Find the speed of \(P\) immediately after receiving the impulse. (5)
| Scheme | Marks | AO |
|---|---|---|
| Impulse momentum equation(s) | M1 | 3.1a |
| \(\begin{pmatrix} 3 \times \cos\alpha \\ 3 \times \sin\alpha \end{pmatrix} = \dfrac{1}{2}\begin{pmatrix} v_x - 2.8 \\ v_y \end{pmatrix}\) \(\left(v_x = \dfrac{32}{5},\ \ v_y = \dfrac{24}{5}\right)\) | A1 A1 | 1.1b 1.1b |
| \(v = \dfrac{1}{5}\sqrt{32^2 + 24^2}\) | M1 | 1.1b |
| \(= 8\ (\text{m s}^{-1})\) | A1 | 1.1b |
| Alternative working parallel and perpendicular to the impulse: \(\begin{pmatrix} 3 \\ 0 \end{pmatrix} = \dfrac{1}{2}\begin{pmatrix} v_1 - 2.8 \times \cos\alpha \\ v_2 \pm 2.8 \times \sin\alpha \end{pmatrix}\) \(v_1 = 7.68,\ v_2 = \pm 2.24\) \(v = \sqrt{7.68^2 + 2.24^2} = 8\ (\text{m s}^{-1})\) | ||
| (5) | ||
| (5 marks) |
Notes
M1: Use of \(\mathbf{I} = m\mathbf{v} - m\mathbf{u}\) in two dimensions. (i.e. resolving used) Dimensionally correct.
Allow for a combined equation in vector format or for just one component. Condone sin/cos confusion.
Allow if \(m\) seen but not substituted.
A1 A1: Equation for one component correct unsimplified
Equations for both components correct unsimplified
Allow A1A1 for a correct unsimplified vector equation
Allow A marks if in terms of \(m\) and \(\alpha\)
M1: Correct use of Pythagoras for their components to obtain the numerical value of the speed
This may be seen or implied: an alert candidate might spot the 3, 4, 5 triangle.
A1: Correct only
Alternative (3alt)
| Scheme | Marks | AO |
|---|---|---|
 | ||
| Using cosine rule: | M1 | |
| \(v^2 = 2.8^2 + 6^2 - 2 \times 2.8 \times 6\cos(\pi - \alpha)\) | A1 A1 | |
| Solve for \(v\) | M1 | |
| \(v = 8\ (\text{m s}^{-1})\) | A1 | |
| (5) |
M1: Correct use of cosine rule in a dimensionally correct triangle. The lengths of the sides must be consistent, i.e. \(v\), 2.8 and 6 or \(\tfrac{1}{2}v\), 1.4 and 3 and it must be a correct vector triangle (vectors combined correctly)
A1 A1: Unsimpified equation with at most one error
Correct unsimplified equation
M1: Substitute for trig. and solve for \(v\)
A1: Correct only