# Flying in the wind

**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.

## Analysis

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

Refer to the above figure. The curve C represents the closed course and is

parameterized by arc length s:

where
represents the length of C. 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 s along the curve

because its direction changes . Finally, the 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 traverse 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 C with no wind at the constant speed p 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 C. 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 v (s) we take the dot product of this equation with itself

which gives a quadratic equation for v(s).

from which we may drop the minus choice which would give a negative value

for v(s).

The equation for t _{wind} becomes

The line integral in the first term is 0 because the
constant vector field
is

conservative. So we have

which completes the argument.

Prev | Next |