4220.21 – Circumnavigating the Earth by Plane

Starting at a point on the equator, suppose a plane sets out due west at 1000 miles per hour. If we let the circumference of the earth be exactly 24,000 miles (it is actually about 24,900 miles), then the sun will not move in the plane’s sky during the flight and it will land at the same hour as it started.

At what degree of latitude would a plane this time flying at 500 mph similarly keep up with the sun in this way?

Solution

The figure below shows the earth and the arc of the two flights. We want the measure of angle $\angle A$ . Note that the radius of the earth, given our assumption about its circumference, is $r = \dfrac{24,000}{2} = 12,000$ .

The second plane will fly a circular route of circumference $500 \text{ mph} \times 24 \text{ hours}$ , that is, 12,000 miles. The radius $y$ of that circle is $y = \dfrac{12,000}{2} = 6,000$ . Thus

$\begin{align*}cos(\angle A) &= \dfrac{y}{r} \\[1em]&= \dfrac{6,000}{12,000} \\[1em]&= \dfrac{1}{2}\end{align*}$

and the angle $\angle A$ has measure 30° latitude.