M2 January 2012 Q2
2. A particle \(P\) is moving in a plane. At time \(t\) seconds, \(P\) is moving with velocity \(\mathbf{v}\) m s\(^{-1}\), where \(\mathbf{v} = 2t\mathbf{i} - 3t^2\mathbf{j}\).
Find
(a) the speed of \(P\) when \(t = 4\) (2)
(b) the acceleration of \(P\) when \(t = 4\) (3)
Given that \(P\) is at the point with position vector \((-4\mathbf{i} + \mathbf{j})\) m when \(t = 1\),
(c) find the position vector of \(P\) when \(t = 4\) (5)
| Scheme | Marks |
|---|---|
| Speed \(= \sqrt{8^2 + 48^2} = \sqrt{2368} = 48.7\ \left(\text{ms}^{-1}\right)\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{a} = 2\mathbf{i} - 6t\mathbf{j}\) | M1 A1 |
| When \(t = 4,\quad \mathbf{a} = 2\mathbf{i} - 24\mathbf{j}\ \left(\text{ms}^{-2}\right)\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(\mathbf{r} = t^2\mathbf{i} - t^3\mathbf{j} + \mathbf{C}\) | M1 A1 |
| \(t = 1,\ -4\mathbf{i} + \mathbf{j} = \mathbf{i} - \mathbf{j} + \mathbf{C},\ \mathbf{C} = -5\mathbf{i} + 2\mathbf{j}\) | DM1 |
| \(\mathbf{r} = (t^2 - 5)\mathbf{i} + \left(-t^3 + 2\right)\mathbf{j}\) | |
| When \(t = 4,\ \mathbf{r} = (16 - 5)\mathbf{i} + (-64 + 2)\mathbf{j} = 11\mathbf{i} - 62\mathbf{j}\) | DM1 A1 |
| (5) | |
| (10 marks) |