M5 June 2006 Q3
3. A particle \(P\) moves in the \(x\)-\(y\) plane and has position vector \(\mathbf{r}\) metres at time \(t\) seconds. It is given that \(\mathbf{r}\) satisfies the differential equation
\[\frac{\mathrm{d}^2\mathbf{r}}{\mathrm{d}t^2} = 2\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}.\]When \(t = 0\), \(P\) is at the point with position vector \(3\mathbf{i}\) metres and is moving with velocity \(\mathbf{j}\) m s\(^{-1}\).
(a) Find \(\mathbf{r}\) in terms of \(t\). (8)
(b) Describe the path of \(P\), giving its cartesian equation. (2)
| Scheme | Marks |
|---|---|
| \(m^2 - 2m = m(m - 2) = 0\) | M1 |
| \(\Rightarrow\ m = 0\) or \(m = 2\) | A1 |
| \(\Rightarrow\ \mathbf{r} = \mathbf{A} + \mathbf{B}\mathrm{e}^{2t}\) | A1 |
| \(t = 0,\ \mathbf{r} = 3\mathbf{i} \;\Rightarrow\; \mathbf{A} + \mathbf{B} = 3\mathbf{i}\) | M1 A1 |
| \(\dot{\mathbf{r}} = 2\mathbf{B}\mathrm{e}^{2t}\) | M1 |
| \(t = 0,\ \dot{\mathbf{r}} = \mathbf{j} \;\Rightarrow\; \mathbf{B} = \tfrac{1}{2}\mathbf{j}\) | A1 |
| \(\Rightarrow\ \mathbf{r} = \left(3\mathbf{i} - \tfrac{1}{2}\mathbf{j}\right) + \tfrac{1}{2}\mathbf{j}\mathrm{e}^{2t} = 3\mathbf{i} + \tfrac{1}{2}\mathbf{j}\left(\mathrm{e}^{2t} - 1\right)\) | A1 |
| (8) |
| Scheme | Marks |
|---|---|
| Particle moves in a straight line | B1 |
| Equation of line is \(x = 3\) | B1 |
| (2) | |
| (10 marks) |