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{\bf Question}

An aeroplane has velocity ${\bf v}_A$ relative to still air.  If
it flies in a wind with velocity ${\bf v}_W$ what is the velocity
of the aeroplane relative to the ground?

\medskip

An aeroplane moves in a northwesterly direction at 500kmh$^{-1}$
due to the fact that there is an easterly wind (i.e. from the
east) of 50kmh$^{-1}$.  Determine how fast and in what direction
the aeroplane would have traveled if there was no wind.

\vspace{.25in}

{\bf Answer}

$\ds {\bf v} = {\bf v}_A + {\bf v}_W$ as ${\bf v}_A$ is relative
to still air; when there is a wind the plane has ${\bf v}_A$
relative to the wind.

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\put(2,4.35){\makebox(0,0){50}}

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\bigskip

cosine rule: \begin{eqnarray*} \ds {\bf v}_A^2 & = & 500^2 + 50^2
- 2 \times 50 \times 500 \cos \frac{\pi}{4} \\ \Rightarrow {\bf
v}_A & = & 466 {\rm kmh}^{-1}\end{eqnarray*}

sine rule:
\begin{eqnarray*} \ds \frac{50}{\sin \alpha} & = & \frac{466}{\sin
\frac{\pi}{4}} \\ \Rightarrow \sin \alpha & = & \frac{25 \sqrt
2}{466} \\ \Rightarrow \alpha & = & 0.076  \textrm{ rads}
\end{eqnarray*}




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