M4 June 2010 Q1
1. [In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively]
A man cycles at a constant speed \(u\) m s\(^{-1}\) on level ground and finds that when his velocity is \(u\mathbf{j}\) m s\(^{-1}\) the velocity of the wind appears to be \(v(3\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\), where \(v\) is a positive constant.
When the man cycles with velocity \(\frac{1}{5}u(-3\mathbf{i} + 4\mathbf{j})\) m s\(^{-1}\), the velocity of the wind appears to be \(w\mathbf{i}\) m s\(^{-1}\), where \(w\) is a positive constant.
Find, in terms of \(u\), the true velocity of the wind. (7)
| Scheme | Marks |
|---|---|
| \(v(3\mathbf{i} - 4\mathbf{j}) = \mathbf{v}_W - u\mathbf{j}\) | M1 A1 |
| \(\mathbf{v}_W = 3v\mathbf{i} + (u - 4v)\mathbf{j}\) | |
| \(w\mathbf{i} = \mathbf{v}_W - \dfrac{u}{5}(-3\mathbf{i} + 4\mathbf{j})\) | M1 A1 |
| \(\mathbf{v}_W = \left(w - \dfrac{3u}{5}\right)\mathbf{i} + \dfrac{4u}{5}\mathbf{j}\) | |
| \((u - 4v) = \dfrac{4u}{5}\) | M1 |
| \(v = \dfrac{u}{20}\) | A1 |
| \(\mathbf{v}_W = \dfrac{3u}{20}\mathbf{i} + \dfrac{4u}{5}\mathbf{j}\) | A1 |
| (7 marks) |