Q: A wind-powered generator converts wind energy into electric energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed v, the electrical power output will be proportional to

(a)v

(b)v^{2}

(c)v^{3}

(d)v^{4}

Ans: (c)

Power P = F v …(i)

$\large F = \frac{dp}{dt} = \frac{d(mv)}{dt}$

$\large F = v \frac{dm}{dt} = v\frac{d(volume \times density)}{dt}$

$\large F = \rho v\frac{dV}{dt}$

$\large F = \rho \times v \times A v$

$\large F = A v^2 \rho$

From(i)

$\large P = A v^3 \rho $

$\large P \propto v^3$