M5 June 2013 (R) Q1
1. A particle moves in a plane in such a way that its position vector \(\mathbf{r}\) metres at time \(t\) seconds satisfies the differential equation
\[\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} - 2\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} = \mathbf{0}\]When \(t = 0\), the particle is at the origin and is moving with velocity \((4\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\).
Find \(\mathbf{r}\) in terms of \(t\). (7)
| Scheme | Marks |
|---|---|
| \(m^2 - 2m = 0\) | M1 |
| \(m = 0\) or 2 | |
| \(\mathbf{r} = \mathbf{A} + \mathbf{B}\mathrm{e}^{2t}\) | A1 |
| \(\dot{\mathbf{r}} = 2\mathbf{B}\mathrm{e}^{2t}\) | M1 A1 |
| \(t = 0,\ \mathbf{r} = \mathbf{0} \Rightarrow \mathbf{A} + \mathbf{B} = \mathbf{0}\) | M1 |
| \(t = 0,\ \dot{\mathbf{r}} = 4\mathbf{i} + 2\mathbf{j} \Rightarrow \mathbf{B} = 2\mathbf{i} + \mathbf{j}\) | |
| \(\Rightarrow \mathbf{A} = -2\mathbf{i} - \mathbf{j}\) | A1 |
| \(\mathbf{r} = (\mathrm{e}^{2t} - 1)(2\mathbf{i} + \mathbf{j})\) | A1 |
| (7) | |
| (7 marks) |