A2 October 2021 Q4
4. A particle \(P\) has mass 0.5 kg. It is moving in the \(xy\) plane with velocity \(8\mathbf{i}\ \text{m s}^{-1}\) when it receives an impulse \(\lambda(-\mathbf{i} + \mathbf{j})\) N s, where \(\lambda\) is a positive constant.
The angle between the direction of motion of \(P\) immediately before receiving the impulse and the direction of motion of \(P\) immediately after receiving the impulse is \(\theta^\circ\)
Immediately after receiving the impulse, \(P\) is moving with speed \(4\sqrt{10}\ \text{m s}^{-1}\)
Find
| Scheme | Marks | AO |
|---|---|---|
| Use of Impulse = change in momentum | M1 | 3.1a |
| \(0.5(\mathbf{v} - 8\mathbf{i}) = \lambda(-\mathbf{i} + \mathbf{j})\) \(\big(\mathbf{v} = (-2\lambda + 8)\mathbf{i} + 2\lambda\mathbf{j}\big)\) | A1 | 1.1b |
| Use of Pythagoras: | M1 | 3.1a |
| e.g. \(160 = (-2\lambda + 8)^2 + (2\lambda)^2\) \((160 = 4\lambda^2 - 32\lambda + 64 + 4\lambda^2)\) | A1 | 1.1b |
| Form and solve quadratic in \(\lambda\): \(8\lambda^2 - 32\lambda - 96 = 0\) \(\big(\lambda^2 - 4\lambda - 12 = (\lambda - 6)(\lambda + 2) = 0\big)\) | M1 | 2.1 |
| \(\Rightarrow \lambda = 6\) | A1 | 1.1b |
Notes
M1: Must be subtracting two values for momentum, but condone subtraction in the wrong order
A1: Correct unsimplified equation
M1: Correct use of final speed with their \(\mathbf{v}\)
A1: Correct unsimplified equation in one unknown or pair of simultaneous equations
M1: Simplify and solve for \(\lambda\) from correct working
A1: Correct positive solution only
| Scheme | Marks | AO |
|---|---|---|
| Find the required angle: \(180^\circ - \tan^{-1}3\) | M1 | 1.1b |
| \(\theta = 108\) | A1 | 2.2a |
| (8) | ||
| (8 marks) |
Notes
M1: Complete method to solve for \(\theta\)
A1: 108 or better (108.4349….)