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Plus Magazine

March 2008

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:
  \[ P \approx 0.38 \times (U/10ms^{-1})^3 \; \;  kilowatts,  \]    
where $U$ is the speed of the approaching wind. We deduced that an average wind speed of $10ms^{-1}$ 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 $20ms^{-1}$ 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.
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