# Maximum height of the water emerging from a sprinkler

The water in a garden hose is at 140 kPA gauge pressure and is moving at negligible speed. The hose terminates in a sprinkler consisting of many small holes. Find the maximum height reached by the water emerging from the holes.

My first thought was to use Bernoulli's equation:

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2$$

We can eliminate a few of these terms: $$\frac{1}{2}\rho v_1^2$$, since the speed in the hose is negligible; $$\rho g y_1$$, since we're measuring the change in height; and $$P_2$$, since this is atmospheric pressure, and the gauge pressure in the hose is measured relative to it. This leaves us with:

$$P_1 = \frac{1}{2}\rho v_2^2 + \rho g y_2$$

This is where I get stuck. I'm trying to solve for $$y_2$$, but $$v_2$$ is unknown, and I can't seem to think of a way to eliminate it. Any help at all that would get me past this point would be very much appreciated. I am a first year physics student, so go easy on me haha.

• Can you use kinematics ($S = ut + 1/2 at^2$) to eliminate one of the two? Commented Nov 18, 2021 at 2:16
• Since you eliminated the speed of water in the hose (at diameter Dh), you eliminated a way to find the speed of water coming out the sprinkler nozzles (at a diameter Ds). Commented Nov 18, 2021 at 2:17

The answer is (drum roll): I totally confused myself.

Using the equation

$$P_1 + \frac{1}{2}\rho v_1^2 + \rho g y_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g y_2$$

Everything on the left has to do with when the fluid is in the hose, and everything on the right has to do with when it's in the sprinkler. Assuming the hose and sprinkler are at the same height, you can eliminate terms to get:

$$P_1 = \frac{1}{2}\rho v_2^2$$

$$v_2 = \sqrt{\frac{2P_1}{\rho}}$$

You can then use kinematics to solve for $$y_{peak}$$:

$$v_{peak}^2 = v_2^2 - 2g(y_{peak} - y_2)$$

$$v_2^2 = 2gy_{peak}$$

$$y_{peak} = \frac{v_2^2}{2g} \rightarrow \frac{\frac{2P_1}{\rho}}{2g}$$

$$\underline{y_{peak} = 14.3 \; m}$$

I hate it when questions have 1 answer that says "never mind, solved," so I figured I'd detail exactly what I did to solve it. Hope this helps.