Flying in the wind
Lynn C. Kurtz, Ph.D.
Arizona State University
Department of Mathematics and Statistics, Retired
An ill wind
You've probably heard the expression "Any wind is an ill wind". When it comes to flying, that isn't true if the wind happens to be a tailwind for your trip. However, it is true if you are talking about flying a closed course. Specifically, if the wind has constant speed and direction and you fly a closed course at a constant airspeed, your best possible time around the course is when the wind is calm.
Description
We will assume that the course the airplane flies can be considered as a simple closed curve in the xy plane and that the speed of the airplane is greater than that of the wind. Otherwise, of course, the plane couldn't negotiate the course. To negotiate a course in the wind, the pilot must "crab" the airplane to account for the effects of the wind and remain on course. Part of the time the wind may be at his tail, increasing his speed over the ground, and part of the time he is heading into the wind, decreasing his ground speed. We want to show that the time taken to traverse the course is least when there is no wind.

NOTE: Some of the inline mathematics symbols do not render and align well on the web. If you want a well typeset version either to read or print, please use the pdf version.

Analysis

We will be using a bit of vector calculus in what follows.
Velocity vectors

Refer to the above figure. The curve represents the closed course and is parameterized by arc length :

where represents the length of . The velocity vector for the airplane is with constant length representing the airplane's speed. The wind is represented by the constant vector with length representing the speed of the wind. Note that while the wind is represented by a constant vector, the airplane's velocity vector depends on its position along the curve because its direction changes. Finally, vector sum of the wind and airplane vectors gives the ground velocity vector whose length represents the speed along the ground. We need to compute the time to complete the course in the wind, which is given by

and show that this time is always at least as long as the time it would take to traverse the curve with no wind at the constant speed of the airplane, which is

The pilot of the airplane continuously adjusts his heading so that the ground velocity vector is tangent to the curve . If we let be the unit vector tangent to the curve, then the ground velocity is

We calculate the ground velocity by adding the wind and airplane velocities:

Solving this equation for gives

To calculate we take the dot product of this equation with itself

which gives a quadratic equation for :

The quadratic formula yields

from which we may drop the minus choice which would give a negative value for . The equation for becomes
The line integral in the first term is zero because the constant vector field is conservative. So we have
which completes the argument.

This document is "copyleft" i.e., not copyrighted 2008. It is for educational purposes and may be freely distributed either electronically or in printed form as long as it is distributed in its entirety. Comments, corrections, and suggestions are welcome. Let me know if you find it useful. My email address is kurtz@asu.edu.