M5 June 2009 Q1
1. At time \(t = 0\), a particle \(P\) of mass 3 kg is at rest at the point \(A\) with position vector \((\mathbf{j} - 3\mathbf{k})\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8\mathbf{i} - 3\mathbf{j} + 5\mathbf{k})\) m.
Given that \(\mathbf{F}_1 = (4\mathbf{i} - 2\mathbf{j} + 5\mathbf{k})\) N and \(\mathbf{F}_2 = (8\mathbf{i} - 4\mathbf{j} + 7\mathbf{k})\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. (7)
| Scheme | Marks |
|---|---|
| \(\pm(8\mathbf{i} - 4\mathbf{j} + 8\mathbf{k})\) | B1 |
| \(\big((4\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}) + (8\mathbf{i} - 4\mathbf{j} + 7\mathbf{k})\big).(8\mathbf{i} - 4\mathbf{j} + 8\mathbf{k}) = \tfrac{1}{2}3v^2\) | M1 A1 f.t. |
| \(12 = v\) | A1 |
| \(\mathbf{v} = \dfrac{12}{\sqrt{8^2 + (-4)^2 + 8^2}}(8\mathbf{i} - 4\mathbf{j} + 8\mathbf{k})\) | M1 |
| \(\mathbf{v} = (8\mathbf{i} - 4\mathbf{j} + 8\mathbf{k})\) m s\(^{-1}\) | DM1 A1 |
| (7 marks) |