October 2021 Paper 3 Mechanics Q1
1. A particle \(P\) moves with constant acceleration \((2\mathbf{i} - 3\mathbf{j})\ \text{m s}^{-2}\)
At time \(t = 0\), \(P\) is moving with velocity \(4\mathbf{i}\ \text{m s}^{-1}\)
At time \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is \((\mathbf{i} + \mathbf{j})\) m.
| Scheme | Marks | AO |
|---|---|---|
| Use of \(\mathbf{v} = \mathbf{u} + \mathbf{a}t\) with \(t = 2\): \(\mathbf{v} = 4\mathbf{i} + 2(2\mathbf{i} - 3\mathbf{j})\) OR integration: \(\mathbf{v} = (2\mathbf{i} - 3\mathbf{j})t + 4\mathbf{i}\), with \(t = 2\) | M1 | 3.1a |
| \(\mathbf{v} = 8\mathbf{i} - 6\mathbf{j}\) | A1 | 1.1b |
| (2) |
Notes
Accept column vectors throughout
M1: Complete method to find \(\mathbf{v}\), using ruva\(t\) or integration
(M0 if \(\mathbf{i}\) and/or \(\mathbf{j}\) is missing)
A1: Apply isw if they also find the speed
| Scheme | Marks | AO |
|---|---|---|
| Use of \(\mathbf{r} = \mathbf{u}t + \dfrac{1}{2}\mathbf{a}t^2\) at \(t = 3\): \((\mathbf{i} + \mathbf{j}) + \left[3 \times 4\mathbf{i} + \dfrac{1}{2} \times (2\mathbf{i} - 3\mathbf{j}) \times 3^2\right]\) OR: find \(\mathbf{v}\) at \(t = 3\): \(4\mathbf{i} + 3(2\mathbf{i} - 3\mathbf{j}) = (10\mathbf{i} - 9\mathbf{j})\) then use \(\mathbf{r} = \dfrac{1}{2}(\mathbf{u} + \mathbf{v})t\) \((\mathbf{i} + \mathbf{j}) + \left[\dfrac{1}{2}\left[4\mathbf{i} + (10\mathbf{i} - 9\mathbf{j})\right] \times 3\right]\) or \(\mathbf{r} = \mathbf{v}t - \dfrac{1}{2}\mathbf{a}t^2\) \((\mathbf{i} + \mathbf{j}) + \left[3 \times (10\mathbf{i} - 9\mathbf{j}) - \dfrac{1}{2} \times (2\mathbf{i} - 3\mathbf{j}) \times 3^2\right]\) OR integration: \(\mathbf{r} = (\mathbf{i} + \mathbf{j}) + \left[(2\mathbf{i} - 3\mathbf{j})\dfrac{1}{2}t^2 + 4t\mathbf{i}\right]\), with \(t = 3\) | M1 | 3.1a |
| \(\mathbf{r} = 22\mathbf{i} - 12.5\mathbf{j}\) | A1 | 2.2a |
| (2) | ||
| (4 marks) |
Notes
Accept column vectors throughout
M1: Complete method to find the p.v. but this mark can be scored if they omit \((\mathbf{i} + \mathbf{j})\)
i.e. the M1 is for the expression in the square bracket
If they integrate, the M1 is earned once the expression in the square bracket is seen with \(t = 3\)
(M0 if \(\mathbf{i}\) and/or \(\mathbf{j}\) is missing)
A1: cao