M2 June 2008 Q4
4. A particle \(P\) of mass 0.5 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds,
\[\mathbf{F} = (6t - 5)\mathbf{i} + (t^2 - 2t)\mathbf{j}.\]The velocity of \(P\) at time \(t\) seconds is \(\mathbf{v}\) m s\(^{-1}\). When \(t = 0\), \(\mathbf{v} = \mathbf{i} - 4\mathbf{j}\).
(a) Find \(\mathbf{v}\) at time \(t\) seconds. (6)
When \(t = 3\), the particle \(P\) receives an impulse \((-5\mathbf{i} + 12\mathbf{j})\) N s.
(b) Find the speed of \(P\) immediately after it receives the impulse. (6)
| Scheme | Marks |
|---|---|
| N2L \((6t - 5)\mathbf{i} + (t^2 - 2t)\mathbf{j} = 0.5\mathbf{a}\) | M1 |
| \(\mathbf{a} = (12t - 10)\mathbf{i} + (2t^2 - 4t)\mathbf{j}\) | A1 |
| \(\mathbf{v} = (6t^2 - 10t)\mathbf{i} + \left(\dfrac{2}{3}t^3 - 2t^2\right)\mathbf{j}\ \ (+\mathbf{C})\) ft their \(\mathbf{a}\) | M1 A1ft+A1ft |
| \(\mathbf{v} = (6t^2 - 10t + 1)\mathbf{i} + \left(\dfrac{2}{3}t^3 - 2t^2 - 4\right)\mathbf{j}\) | A1 |
| (6) |
| Scheme | Marks |
|---|---|
| When \(t = 3\), \(\mathbf{v}_3 = 25\mathbf{i} - 4\mathbf{j}\) | M1 |
| \(-5\mathbf{i} + 12\mathbf{j} = 0.5\left(\mathbf{v} - (25\mathbf{i} - 4\mathbf{j})\right)\) ft their \(\mathbf{v}_3\) | M1 A1ft |
| \(\mathbf{v} = 15\mathbf{i} + 20\mathbf{j}\) | A1 |
| \(|\mathbf{v}| = \sqrt{15^2 + 20^2} = 25\) (m s\(^{-1}\)) cso | M1 A1 |
| (6) | |
| (12 marks) |