Worked example 3.5: Flight UA589

Question: United Airlines flight UA589 from Chicago is 20miles due North of Austin's Bergstrom airport. Suppose that the plane is flying at $200 {\rm mi/h}$ relative to the air. Suppose, further, that there is a wind blowing due East at $60 {\rm mi/h}$. Towards which compass bearing must the plane steer in order to land at the airport?
 
Answer: The problem in hand is illustrated in the diagram.

The plane's velocity ${\bf v}_g$ relative to the ground is the vector sum of its velocity ${\bf v}_a$ relative to the air, and the velocity ${\bf u}$ of the wind relative to the ground. We know that ${\bf u}$ is directed due East, and we require ${\bf v}_g$ to be directed due South. We also know that $\vert{\bf v}_a\vert=200 {\rm mi/h}$ and $\vert{\bf u}\vert = 60 {\rm mi/h}$. Now, from simple trigonometry,

$$\cos\alpha = \frac{\vert{\bf u}\vert}{\vert{\bf v}_a\vert} = \frac{60}{200} =0.3.$$

Hence,

$$\alpha = 72.54^\circ.$$

However, it is clear from the diagram that the compass bearing $\phi$ of the plane is given by

$$\phi = 270^\circ -alpha = 270^\circ - 72.54^\circ = 197.46^\circ.$$

Thus, in order to land at Bergstrom airport the plane must fly towards compass bearing $197.46^\circ$.