Q103 / 197 · NEET 2009 (Mains) · Physics

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 ?

  • A
    $\frac{1}{2} m^2 v^2$
  • Correct answer
    B
    $\frac{1}{2} \mathrm{mv}^3$
  • C
    $\mathrm{mv}^3$
  • D
    $\frac{1}{2} \mathrm{mv}^2$

Solution

$\mathrm{mv}=\frac{\mathrm{dm}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dt}}=\frac{\mathrm{dm}}{\mathrm{dt}}=$ Rate of flowing mass $$ \begin{aligned} & \mathrm{F}_{\mathrm{av}}=\frac{\mathrm{dm}}{\mathrm{dt}} \times \frac{\mathrm{v}}{2}=\frac{(\mathrm{mv}) \mathrm{v}}{2}=\frac{\mathrm{mv}^2}{2} \\ & \mathrm{p}=\frac{\mathrm{dK}}{\mathrm{dt}}=\frac{\mathrm{mv}^2}{2} \times \mathrm{v}=\frac{\mathrm{mv}^3}{3} \end{aligned} $$

Hence, the correct answer is option: 'B'.

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