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Question

An engine pumps water continuously through a hole. Speed with which water passes through the hole nozzle is v, and k is the mass per unit length of the water jet as it leaves the nozzle. Find the rate at which kinetic energy is being imparted to the water.

Moderate

A

$\frac{1}{2} k v^{2}$
B

$\frac{1}{2} k v^{3}$
C

$\frac{v^{2}}{2 k}$
D

$\frac{v^{3}}{2 k}$

Solution

mass per unit length of water jet is K = $\frac{M a s s}{L e n g t h} = \frac{d m}{d x},$
v = speed of water
Kinetic energyKE = $\frac{1}{2} m v^{2}$
differentiate with respect to time,
$\frac{d}{d t}$ (KE) = $\frac{1}{2}$ $(\frac{d m}{d t})$ $v^{2}$
divide and multiply with dx,
$\frac{d}{d t}$ (KE) = $\frac{1}{2} (\frac{d m}{d x} . \frac{d x}{d t})$ $v^{2}$
rate at which kinetic energy is being imparted to the water = $\frac{k v v^{2}}{2}$ = $\frac{k v^{3}}{2}$

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