M2 January 2008 Q2
2. At time \(t\) seconds \((t \geqslant 0)\), a particle \(P\) has position vector \(\mathbf{p}\) metres, with respect to a fixed origin \(O\), where
\[\mathbf{p} = (3t^2 - 6t + 4)\mathbf{i} + (3t^3 - 4t)\mathbf{j}.\]Find
(a) the velocity of \(P\) at time \(t\) seconds, (2)
(b) the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf{i}\). (3)
When \(t = 1\), the particle \(P\) receives an impulse of \((2\mathbf{i} - 6\mathbf{j})\) N s. Given that the mass of \(P\) is 0.5 kg,
(c) find the velocity of \(P\) immediately after the impulse. (4)
| Scheme | Marks |
|---|---|
| \(\dot{\mathbf{p}} = (6t - 6)\mathbf{i} + (9t^2 - 4)\mathbf{j}\) (m s\(^{-1}\)) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(9t^2 - 4 = 0\) | M1 |
| \(t = \tfrac{2}{3}\) | DM1 A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(t = 1 \ \Rightarrow\ \dot{\mathbf{p}} = 5\mathbf{j}\) ft their \(\dot{\mathbf{p}}\) | B1ft |
| (+/-) \(2\mathbf{i} - 6\mathbf{j} = 0.5(\mathbf{v} - 5\mathbf{j})\) | M1 |
| \(\mathbf{v} = 4\mathbf{i} - 7\mathbf{j}\) (m s\(^{-1}\)) | M1 A1 |
| (4) | |
| (9 marks) |