M2 January 2006 Q2
2. A particle \(P\) of mass 0.4 kg is moving so that its position vector \(\mathbf{r}\) metres at time \(t\) seconds is given by
\[\mathbf{r} = (t^2 + 4t)\mathbf{i} + (3t - t^3)\mathbf{j}.\](a) Calculate the speed of \(P\) when \(t = 3\). (5)
When \(t = 3\), the particle \(P\) is given an impulse \((8\mathbf{i} - 12\mathbf{j})\) N s.
(b) Find the velocity of \(P\) immediately after the impulse. (3)
| Scheme | Marks |
|---|---|
| \(\dot{\mathbf{r}} = (2t + 4)\mathbf{i} + (3 - 3t^2)\mathbf{j}\) | M1 A1 |
| \(\dot{\mathbf{r}}_3 = 10\mathbf{i} - 24\mathbf{j}\) substituting \(t = 3\) | M1 |
| \(|\dot{\mathbf{r}}_3| = \sqrt{(10^2 + 24^2)} = 26\ \ (\text{m s}^{-1})\) | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(0.4\big(\mathbf{v} - (10\mathbf{i} - 24\mathbf{j})\big) = 8\mathbf{i} - 12\mathbf{j}\) ft their \(\dot{\mathbf{r}}_3\) | M1 A1ft |
| \(\mathbf{v} = 30\mathbf{i} - 54\mathbf{j}\ \ (\text{m s}^{-1})\) | A1 |
| (3) | |
| (8 marks) |