# Question on power

A wind turbine produces a power $P$ when the wind speed is $v$. Assuming that the efficiency of the turbine is constant, the best estimate for the power produced when the wind speed becomes $2v$ is

(1) $2P$

(2) $4P$

(3) $6P$

(4) $8P$

My doubt here is that power, $P= F.v$ and basically if i write $F=$ $m { d v\over dt}$ in this expression i get a dependance of $P$ directly proportional to $v^2$ which means the answer should be $4P$ but the answer comes to $8P$. Why is that?? What other relations can be used in linking power to velocity? Thanks in advance.

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Consider the energy being transferred, $$E = \frac{1}{2}mv^2$$
At $2v$, the mass of air impinging on the turbine also doubles, so effectively
$$E\propto v^3$$ as $m = \rho V = \rho Avt$, where $\rho$ = air density, $A$ = area, and $t$ = time
Your reasoning is ok if you consider the same mass of air going through the turbine at different speeds. But when the speed is doubled you also have twice air going through in the same time interval, that's where the extra factor of $2$ comes from.