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)v2
(c)v3
(d)v4
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 $
$\large P \propto v^3$