Blowin' in the wind: solution

In last issue's Outer space we developed a formula for the power output per unit time of a windmill:$$P\approx 0.38\times (U/10m{s}^{-1}{)}^{3}kilowatts,$$ where $U$ is the speed of the approaching wind. We deduced that an average wind speed of $10m{s}^{-1}$ gives a power output approximately equal to 0.38. The question was why this calculation significantly underestimates the true output. \ The answer lies in understanding that the average speed is not a good guide. If, for example, the wind speed was $20m{s}^{-1}$ for half of the time unit and 0 for the other half, then we would get an output approximately equal to $$\frac{1}{2}(0.38\times 2^3)=4\times 0.38.$$ This is four times more than what we got in the original calculation. Back to Outer space