How fast do you need to travel downwind to obtain the maximum amount of power from the wind?

Question

Suppose you are traveling downwind, in a wind speed of $x$ meters per second. If you are stationary, the wind speed relative to you will be the highest, but you will not be moving, so no work/power is done, because work is force times distance, and power is work over time.

If you are traveling at $x$ meters per second, you travel a distance of $x$ meters every second but since you are traveling at the exact same speed as the wind, the wind will impart no force on you.

Outside of these points, where your downwind speed is negative (traveling upwind) or greater than the windspeed, you will be imparting work on the air, so the power you receive is negative.

Somewhere in between a speed of 0 and $x$, the wind will impart a force on you while you are moving, amounting to work/power. At what speed will the power be the greatest? At what fraction of the wind speed would you receive the most power from the wind?

Answer 1

It is simple enough to perform the derivation ourselves if we make some basic assumptions. Namely, let us take the drag force to be quadratic in the relative speed. For a wind speed $u$, and object speed $v$, then we can write the power imparted by drag: $$P=Fv=C(u-v)^2v$$ where $C$ is some constant which depends on the density of the fluid, the projected area in the direction of motion, etc.

Now we just need to maximize $P$ in terms of $v$:

$$\begin{align*} P(v)&=C(u-v)^2v\\ \mathrm{d}_vP&=C[(u-v)^2-2v(u-v)]=0\\ &=(u-v)(u-3v)\\ u=3v\implies v&=\boxed{u/3} \end{align*}$$

So, we should expect maximum power at one third of the wind speed.