# Math Help - Extremely difficult question on relative velocity?

1. ## Extremely difficult question on relative velocity?

A man is walking in the north-east direction and wind appears to blow from north. If the man doubles his speed, wind appears at angle arccot 2 east of north. Find the actual direction of the wind.

2. The wind comes from the west. You can check that it works; anyway, here is how I found this out:

Let us denote by $\vec{w}={w_E\choose w_N}$ and $\vec{v}={\alpha \choose \alpha}$ the respective speeds of the wind and the man, in an orthonormal basis $(\vec{e}_E,\vec{e}_N)$ (E for east, N for north). Since the man goes NE, both components of $v$ are equal to some positive $\alpha$.

We know that the relative speed, i.e. the vector $\vec{w}-\vec{v}={w_E-\alpha\choose w_N-\alpha}$, is along the N direction, hence the E component is zero: $w_E=\alpha$.

Then we are told that the vector $\vec{w}-2\vec{v}={w_E-2\alpha\choose w_N-2\alpha}$ is (positively) colinear with $-{1\choose 2}$. This is the content of the statement about "arccot 2" (draw a right triangle with perpendicular sides of length 1 and 2; the least angle measures ${\mathop{arccot}}(2)$). As a consequence, $\frac{w_N-2\alpha}{w_E-2\alpha}=2$. Replace $\alpha$ by $w_E$ (as seen above), to get $2-\frac{w_N}{w_E}=2$, hence $w_N=0$.

Finally, $\vec{w}={\alpha\choose 0}$ (the same $\alpha$ as in $\vec{v}$), and $\alpha>0$, which means that the wind comes from the west.

I hope my explanations are clear.
Laurent.