An engine pumps a liquid of density d continu | vedclass

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(A) The mass of the liquid passing through the pipe per unit time is given by the rate of mass flow: $\frac{dm}{dt} = \rho A V$,where $\rho = d$ is th

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$\frac{1}{2} AdV^3$

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(A) The mass of the liquid passing through the pipe per unit time is given by the rate of mass flow: $\frac{dm}{dt} = \rho A V$,where $\rho = d$ is the density.Thus,$\frac{dm}{dt} = dAV$.The kinetic energy $K$ of a mass $m$ moving with speed $V$ is $K = \frac{1}{2} m V^2$.The kinetic energy imparted per unit time is $\frac{dK}{dt} = \frac{1}{2} V^2 \frac{dm}{dt}$.Substituting the value of $\frac{dm}{dt}$,we get $\frac{dK}{dt} = \frac{1}{2} V^2 (dAV) = \frac{1}{2} AdV^3$.

An engine pumps a liquid of density $d$ continuously through a pipe of cross-sectional area $A$. If the speed of the liquid passing through the pipe is $V$,what is the kinetic energy per unit time imparted to the liquid?

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