M1 January 2007 Q3
3. A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find
(a) the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), (2)
(b) the magnitude of \(\mathbf{F}\), (4)
(c) the velocity of \(P\) at time \(t = 6\) s. (3)
| Scheme | Marks |
|---|---|
| \(\mathbf{a} = \dfrac{(15\mathbf{i} - 4\mathbf{j}) - (3\mathbf{i} + 2\mathbf{j})}{4} = 3\mathbf{i} - 1.5\mathbf{j}\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| N2L \(\mathbf{F} = m\mathbf{a} = 6\mathbf{i} - 3\mathbf{j}\) ft their \(\mathbf{a}\) | M1 A1 |
| \(|\mathbf{F}| = \sqrt{(6^2 + 3^2)} \approx 6.71\) (N) accept \(\sqrt{45}\), awrt 6.7 | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathbf{v}_6 = (3\mathbf{i} + 2\mathbf{j}) + (3\mathbf{i} - 1.5\mathbf{j})6\) ft their \(\mathbf{a}\) | M1 A1ft |
| \(= 21\mathbf{i} - 7\mathbf{j}\) (m s\(^{-1}\)) | A1 |
| (3) | |
| (9 marks) |