An engine pumps water continuously through a hose. Water leaves the hose with a velocity $v$ and $m$ is the ma

Question

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

Accepted Answer

1. **Identify the Goal:** We need to find the rate at which kinetic energy is imparted, which is the Power ($P$).
2. **Formula for Power:** Power is the rate of change of kinetic energy: $P = \frac{dK}{dt}$.
3. **Kinetic Energy Formula:** The kinetic energy of a mass $M$ moving with velocity $v$ is $K = \frac{1}{2} M v^2$ .
4. **Differentiate:** Since $v$ is constant, the rate of change is: $\frac{dK}{dt} = \frac{1}{2} v^2 \frac{dM}{dt}$.
5. **Determine Mass Flow Rate:** We are given $m$ as the mass per unit length (linear density). The mass $dM$ flowing in time $dt$ corresponds to a length $dx = v dt$. Thus, $dM = m dx = m (v dt)$. The mass flow rate is $\frac{dM}{dt} = mv$ .
6. **Substitute and Solve:**
   $$ P = \frac{1}{2} v^2 (mv) = \frac{1}{2} m v^3 $$

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