An engine pumps water continuously through a hose. Water leaves the hose with a velocity $v$ and $\lambda$ is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to the water?

Solution

1. The mass flow rate of the water jet is given by the product of the mass per unit length and the velocity: $\frac{dm}{dt}=\lambda v$.
2. The kinetic energy of a mass $m$ moving at velocity $v$ is $K=\frac{1}{2}m{v}^2$.
3. The rate at which kinetic energy is imparted is the time derivative of $K$, given by $$\begin{array}{rl}P& =\frac{dK}{dt}\\ & =\frac{1}{2}{v}^2\frac{dm}{dt}\end{array}$$.
4. Substituting the mass flow rate into the power equation yields $$\begin{array}{rl}P& =\frac{1}{2}{v}^2(\lambda v)\\ & =0.5\lambda v^3\end{array}$$.
5. Hence the answer is (A).