M5 June 2008 Q1
1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.]
A small bead of mass 0.5 kg is threaded on a smooth horizontal wire. The bead is initially at rest at the point with position vector \((\mathbf{i} - 6\mathbf{j})\) m. A constant horizontal force \(\mathbf{P}\) N then acts on the bead causing it to move along the wire. The bead passes through the point with position vector \((7\mathbf{i} - 14\mathbf{j})\) m with speed \(2\sqrt{7}\) m s\(^{-1}\).
Given that \(\mathbf{P}\) is parallel to \((6\mathbf{i} + \mathbf{j})\), find \(\mathbf{P}\).
| Scheme | Marks |
|---|---|
| \(\mathbf{d} = (7\mathbf{i} - 14\mathbf{j}) - (\mathbf{i} - 6\mathbf{j}) = (6\mathbf{i} - 8\mathbf{j})\) | B1 |
| \((6k\mathbf{i} + k\mathbf{j}).(6\mathbf{i} - 8\mathbf{j}) = \tfrac{1}{2}\times\tfrac{1}{2}\times\left(2\sqrt{7}\right)^2\) | M1 A2 ft |
| \(28k = 7 \Rightarrow k = \tfrac{1}{4}\) | DM1 |
| \(\Rightarrow\ \mathbf{P} = \tfrac{3}{2}\mathbf{i} + \tfrac{1}{4}\mathbf{j}\) | A1 |
| (6 marks) |