Q. 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
A) $\frac{1}{2}kv^{2}$
B) $\frac{1}{2}kv^{3}$
C) $\frac{v^{2}}{2k}$
D) $\frac{v^{3}}{2k}$
Explanation
(b) $K = \frac{mass}{length} = \frac{dm}{dt}$
$KE = \ \frac{1}{2}mv^{2}\ \Rightarrow \ \frac{d}{dt}(KE) = \frac{1}{2}\left( \frac{dm}{dt} \right)v^{2}$
$= \frac{1}{2}\left( \frac{dm}{dx} \times \frac{dx}{dt} \right)v^{2}$
$= \frac{1}{2}kvv^{2} = \frac{1}{2}kv^{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.