A2 June 2025 Q3
3. A particle \(P\) of mass 0.4 kg is moving with velocity \(7\mathbf{i}\ \text{m s}^{-1}\) when it receives an impulse of magnitude \(\sqrt{1.6}\ \text{N s}\).
The velocity of \(P\) immediately after it receives the impulse is \(\lambda(2\mathbf{i} + \mathbf{j})\ \text{m s}^{-1}\), where \(\lambda\) is a constant.
Find the two possible values of \(\lambda\) (6)
| Scheme | Marks | AO |
|---|---|---|
| Use of change in momentum | M1 | 3.1a |
| \((\mathbf{I}) = 0.4\lambda(2\mathbf{i} + \mathbf{j}) - 0.4 \times 7\mathbf{i}\) | A1 | 1.1b |
| Use of Pythagoras to form an equation for magnitude of impulse | M1 | 1.1b |
| E.g. \(1.6 = (0.8\lambda - 2.8)^2 + (0.4\lambda)^2\) \(1.6 = 0.4^2\left((2\lambda - 7)^2 + \lambda^2\right)\) | A1 | 1.1b |
| \(5\lambda^2 - 28\lambda + 39 = 0 \quad\Rightarrow\quad \lambda = \ldots\) | dM1 | 2.1 |
| \(\lambda = 2.6,\quad \lambda = 3\) | A1 | 2.2a |
| (6) | ||
| (6 marks) |
Notes
Accept column vectors throughout
M1: Form an expression for change in momentum in \(\lambda\), correct number of terms and dimensionally correct. Must use velocities and both components. Subtraction may be either way round. If present, ignore LHS. For the M mark, condone poor expanding of the velocity i.e. \((0.8\lambda\mathbf{i} + 0.4\mathbf{j})\) or \((0.8\mathbf{i} + 0.4\lambda\mathbf{j})\)
M0 if speed is used
A1: Correct unsimplified expression for change in momentum, accept terms either way round. Must use conventional vector notation ie column vector form or i-j form.
If present, ignore LHS.
A0 for unconventional vector notation, unless recovered.
M1: Correct use of Pythagoras (squaring and adding) to form an equation for magnitude of impulse. Must use the given magnitude, \(\sqrt{1.6}\), and the change in momentum components, to form an equation in \(\lambda\) only. Other unknowns may be introduced to represent the impulse components. E.g. \(\begin{pmatrix} a \\ b \end{pmatrix} = 0.4\begin{pmatrix} 2\lambda \\ \lambda \end{pmatrix} - 0.4\begin{pmatrix} 7 \\ 0 \end{pmatrix}\)
However, the M mark is only awarded when Pythagoras is used correctly to form an equation in \(\lambda\) only. E.g. \(\left(\sqrt{1.6}\right)^2 = a^2 + b^2 \;\Rightarrow\; 1.6 = (0.8\lambda - 2.8)^2 + (0.4\lambda)^2\)
A1: Correct unsimplified equation in \(\lambda\) only.
dM1: Dependent on previous two M’s. Complete method using the change in momentum and magnitude of impulse to form a 3TQ in \(\lambda\) only and solve to find two \(\lambda\) values. No need to see the method for solving 3TQ. Must reach \(\lambda = \ldots\)
A1: Both correct values o.e. eg \(\lambda = \dfrac{13}{5},\ \lambda = 3\)