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

Starting from the definition of work done by a force, $W=\int \vec F\cdot d\vec r$, and differentiating with respect to time (using $d\vec r/dt=\vec v$, the instantaneous velocity of the body), gives the instantaneous power delivered by that force as the dot product of the force and the velocity:

This relation is the standard tool for computing the power an engine or motor must supply. For a vehicle moving at speed $v$ against a resistive force $F_{\text{resistive}}$ (air drag, rolling friction) while also accelerating at $a$, the engine's total driving force must equal $F_{\text{resistive}}+ma$ (Newton's second law), and the instantaneous power required is this total force multiplied by the current speed, $P=(F_{\text{resistive}}+ma)\,v$. The same relation explains why a wind turb …