An engine pumps water continuously through a hose. Water leaves the hose with a velocity $\mathrm{v}$ and…
An engine pumps water continuously through a hose. Water leaves the hose with a velocity $\mathrm{v}$ and $\mathrm{m}$ is the mass per unit length of the water jet. What is the rate which kinetic energy is imparted to water ?
$\frac{1}{2} m^2 v^2$
$\frac{1}{2} \mathrm{mv}^3$
$\mathrm{mv}^3$
$\frac{1}{2} \mathrm{mv}^2$
Solution
$\mathrm{mv}=\frac{\mathrm{dm}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dt}}=\frac{\mathrm{dm}}{\mathrm{dt}}=$ Rate of flowing mass
$\begin{aligned}
& \mathrm{F}_{\mathrm{av}}=\frac{\mathrm{dm}}{\mathrm{dt}} \times \frac{\mathrm{v}}{2}=\frac{(\mathrm{mv}) \mathrm{v}}{2}=\frac{\mathrm{mv}^2}{2} \\
& \mathrm{p}=\frac{\mathrm{dK}}{\mathrm{dt}}=\frac{\mathrm{mv}^2}{2} \times \mathrm{v}=\frac{\mathrm{mv}^3}{2}
\end{aligned}$