$A$ 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 blades of a windmill sweep out a circle of area $A$.
$(a)$ If the wind flows at a velocity $v$ perpendicular to the circle,what is the mass of the air passing through it in time $t?$
$(b)$ What is the kinetic energy of the air?
$(c)$ Assume that the windmill converts $25\%$ of the wind's energy into electrical energy,and that $A=30\;m^{2}, v=36\;km/h$ and the density of air is $1.2\;kg\;m^{-3}.$ What is the electrical power produced?

An engine pumps $100 \ kg$ of water to a height of $10 \ m$ in $5 \ s$. If the efficiency of the engine is $60\%$,the power of the engine is........$kW$. (Take $g = 10 \ m/s^2$.)

The power of a pump,which can pump $500 \,kg$ of water to a height of $100 \,m$ in $10 \,s$,is .......... $kW$. (Take $g = 10 \,m/s^2$)

$A$ body of mass $4 \ kg$ is placed on a plane at a point $P$ having coordinates $(3, 4) \ m$. Under the action of force $\overrightarrow{F} = (2 \hat{i} + 3 \hat{j}) \ N$,it moves to a new point $Q$ having coordinates $(6, 10) \ m$ in $4 \ s$. The average power and instantaneous power at the end of $4 \ s$ are in the ratio of:

$A$ body moves along a straight line under the action of a constant power delivered by an engine. The distance covered by the body in time $t$ is proportional to: