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  
\[\mathbf{w}_{TRUE}=\mathbf{v}_{CYCLIST}+ \mathbf{w}_{APPARENT}\]
.

true wind speed

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

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