## Finding True Wind Speed With a Change of Direction

A man cycles at a constant velocity
$u \mathbf{j}$
m/s on level ground and finds the apparent velocity of the wind is
$v(3 \mathbf{i} - 4 \mathbf{j} )$
m/s.
When he cycles with velocity
$\frac{u}{5}(-3 \mathbf{i} + 4 \mathbf{j} )$
m/s, the apparent velocity of the wind is
$w \mathbf{i}$
m/s.
What is the true wind velocity?
The diagram below illustrates that the true wind speed is
$\mathbf{w}_{TRUE}=\mathbf{v}_{CYCLIST}+ \mathbf{w}_{APPARENT}$
.

From the question then
$\mathbf{w}_{TRUE}=u \mathbf{j}+v(3 \mathbf{i} - 4 \mathbf{j} )=3 v \mathbf{i} +(u- 4v) \mathbf{j}$

$\mathbf{w}_{TRUE}=\frac{v}{5}(-3 \mathbf{i} + 4 \mathbf{j} )+w \mathbf{i}=(- \frac{3v}{5}+w) \mathbf{i} - \frac{4v}{5} \mathbf{j}$

Equating these
$3 v \mathbf{i} +(u- 4v) \mathbf{j}=(- \frac{3v}{5}+w) \mathbf{i} - \frac{4v}{5} \mathbf{j}$

Equating components:
$3 v=(- \frac{3v}{5}+w)$
(1)
$u- 4v = - \frac{4v}{5}v$
(2)
From (1)
$3.6 v=w$
and from (2)
$u=4.8v$
so
$w=\frac{3.6}{4.8}u=\frac{3}{4}u$
.
$\mathbf{w}_{TRUE}=(- \frac{3v}{5}+w) \mathbf{i} - \frac{4v}{5} \mathbf{j}=(- \frac{3u/4.8}{5}+\frac{3}{4}u) \mathbf{i} - \frac{4u/4.8}{5} \mathbf{j}=- \frac{u}{6} \mathbf{j}$

Equating these