A wind-powered generator converts wind energy into electrical 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 most likely proportional to

The wind-powered generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. To find how the electrical power output depends on wind speed $$ v $$, we need to determine the power available from the wind.

The kinetic energy of a mass $$ m $$ of air moving at speed $$ v $$ is given by $$ \frac{1}{2} m v^2 $$. Power is energy per unit time, so the power from the wind is the rate at which this kinetic energy is delivered to the blades. This can be expressed as:

$$ P_{\text{wind}} = \frac{d}{dt} \left( \frac{1}{2} m v^2 \right) $$

Since the wind speed $$ v $$ is constant, we can factor it out:

$$ P_{\text{wind}} = \frac{1}{2} v^2 \frac{dm}{dt} $$

Here, $$ \frac{dm}{dt} $$ is the mass flow rate of air passing through the blades per unit time. The mass flow rate depends on the density of air $$ \rho $$, the area $$ A $$ swept by the blades, and the wind speed $$ v $$. The volume of air passing through area $$ A $$ per unit time is $$ A v $$, so the mass flow rate is:

$$ \frac{dm}{dt} = \rho \times A v $$

Substituting this into the power expression:

$$ P_{\text{wind}} = \frac{1}{2} v^2 \times (\rho A v) = \frac{1}{2} \rho A v^3 $$

Thus, the power available from the wind is proportional to $$ v^3 $$.

The problem states that the generator converts a fixed fraction of this intercepted wind energy into electrical energy. Let this fixed fraction (efficiency) be $$ \eta $$. The electrical power output is then:

$$ P_{\text{electrical}} = \eta \times P_{\text{wind}} = \eta \times \frac{1}{2} \rho A v^3 $$

Since $$ \eta $$, $$ \rho $$, and $$ A $$ are constants for a given generator and location, the electrical power output is proportional to $$ v^3 $$.

Therefore, the electrical power output is most likely proportional to $$ v^3 $$. Comparing with the options:

A. $$ v^4 $$

B. $$ v^2 $$

C. $$ v $$

D. $$ v^3 $$

Hence, the correct answer is Option D.