M5 June 2006 Q2
2. A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\).
| Scheme | Marks |
|---|---|
| \(\mathbf{d} = \begin{pmatrix}4\\5\\-5\end{pmatrix} - \begin{pmatrix}2\\3\\-4\end{pmatrix} = 2\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) | B1 |
| \(\mathbf{F}\cdot(2\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = \tfrac{1}{2}\times\tfrac{1}{2}\times 12^2 = 36\) | M1 A2 |
| but \(\mathbf{F} = \lambda(2\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) (particle starts from rest) | M1 |
| \(\Rightarrow\ \lambda(2\mathbf{i} + 2\mathbf{j} - \mathbf{k})\cdot(2\mathbf{i} + 2\mathbf{j} - \mathbf{k}) = 36\) | M1 |
| \(\Rightarrow\ 9\lambda = 36\) | |
| \(\Rightarrow\ \lambda = 4\) | A1 |
| \(\mathbf{F}_2 = 4\begin{pmatrix}2\\2\\-1\end{pmatrix} - \begin{pmatrix}1\\2\\-1\end{pmatrix} = 7\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}\) | M1 A1 |
| (9 marks) |