A wind mill converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. The electrical power output P is related to the velocity of wind v as

Step-by-step solution

The kinetic energy K of a mass m of air moving with velocity v is given by K = 1 2 mv^2 . The mass m of air passing through an area A in time t is m = A v t , where is the density of air. The power P is the rate of change of energy, P = K t = 1 2 ( m t ) v^2 . Substituting the expression for mass flow rate m t = A v , we get P = 1 2 ( A v) v^2 = 1 2 A v^3 . Since the windmill converts a fixed fraction of this energy, the electrical power output P is directly proportional to v^3 . Answer: v^3