M2 January 2007 Q6
6. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds, \(\mathbf{F} = (1.5t^2 - 3)\mathbf{i} + 2t\mathbf{j}\). When \(t = 2\), the velocity of \(P\) is \((-4\mathbf{i} + 5\mathbf{j})\) m s\(^{-1}\).
(a) Find the acceleration of \(P\) at time \(t\) seconds. (2)
(b) Show that, when \(t = 3\), the velocity of \(P\) is \((9\mathbf{i} + 15\mathbf{j})\) m s\(^{-1}\). (5)
When \(t = 3\), the particle \(P\) receives an impulse \(\mathbf{Q}\) N s. Immediately after the impulse the velocity of \(P\) is \((-3\mathbf{i} + 20\mathbf{j})\) m s\(^{-1}\). Find
(c) the magnitude of \(\mathbf{Q}\), (3)
(d) the angle between \(\mathbf{Q}\) and \(\mathbf{i}\). (3)
| Scheme | Marks |
|---|---|
| N2L \((1.5t^2 - 3)\mathbf{i} + 2t\mathbf{j} = 0.5\mathbf{a}\) | M1 |
| \(\mathbf{a} = (3t^2 - 6)\mathbf{i} + 4t\mathbf{j}\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{v} = (t^3 - 6t)\mathbf{i} + 2t^2\mathbf{j}\ \ (+\mathbf{c})\) | M1 A1 |
| \(t = 2\) \(-4\mathbf{i} + 5\mathbf{j} = -4\mathbf{i} + 8\mathbf{j} + \mathbf{c}\) \((\mathbf{c} = -3\mathbf{j})\) | M1 |
| \(\mathbf{v} = (t^3 - 6t)\mathbf{i} + (2t^2 - 3)\mathbf{j}\ \ (\text{m s}^{-1})\) | A1 |
| \(t = 3\) \(\mathbf{v} = 9\mathbf{i} + 15\mathbf{j}\ (\text{m s}^{-1})\ \ *\) cso | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(\mathbf{Q} = 0.5\big(-3\mathbf{i} + 20\mathbf{j} - (9\mathbf{i} + 15\mathbf{j})\big)\ \ \big(= 0.5(-12\mathbf{i} + 5\mathbf{j})\big)\) | M1 |
| \(|\mathbf{Q}| = 0.5\sqrt{(5^2 + 12^2)} = 6.5\) | M1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| acute angle is \(\arctan\dfrac{5}{12} \approx 23^\circ\) or required angle is \(\arctan\dfrac{-5}{12}\) or acute angle is \(\arccos\dfrac{12}{13} \approx 23^\circ\) or required angle is \(\arccos\dfrac{-12}{13}\) | M1 A1 |
| required angle is 157\(^\circ\) awrt 157\(^\circ\), 203\(^\circ\) | A1 |
| (3) | |
| (13 marks) |