Question

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 proportional to

• A. $V$
• B. $V^2$
• C. $V^3$
• D. $V^4$

Answer 1

The correct option is C $V^3$

Number of particles striking the blades/time $\propto$ velocity of wind
KE of $\propto (velocityofwind{)}^2$
$\therefore$ Power output $\propto$ (Number of particles striking/time) \times (KE of particle) $\propto V^3$
Alternative method :
$F=v\left(\frac{dm}{dt}\right)=v\frac{d}{dt}(\rho \times volume)$
$=v\rho \frac{d}{dt}\left(volume\right)=v\rho \times \left(Av\right)=A\rho v^2$
$Power=Force\times Velocity$
$=A\rho v^2\times v=A\rho v^3\Rightarrow P\propto v^3$