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(A) The kinetic energy $K$ of a mass $m$ moving with velocity $v$ is given by $K = \frac{1}{2}mv^2$.The rate at which kinetic energy is imparted is th

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$\frac{1}{2}A\rho v^3$

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(A) The kinetic energy $K$ of a mass $m$ moving with velocity $v$ is given by $K = \frac{1}{2}mv^2$.The rate at which kinetic energy is imparted is the power $P = \frac{dK}{dt} = \frac{1}{2}v^2 \frac{dm}{dt}$.The mass $m$ flowing through the pipe in time $t$ is $m = 	ext{Volume} 	imes 	ext{density} = (A 	imes l) 	imes ho$,where $l$ is the length of the liquid column.Thus,the rate of mass flow is $\frac{dm}{dt} = A 	imes \frac{dl}{dt} 	imes ho = A v ho$.Substituting this into the power equation: $P = \frac{1}{2}v^2 (A v ho) = \frac{1}{2}A ho v^3$.

An engine pump is used to pump a liquid of density $\rho$ continuously through a pipe of cross-sectional area $A$. If the speed of flow of the liquid in the pipe is $v$,then the rate at which kinetic energy is being imparted to the liquid is

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