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A water tank has two pipes installed on the outside near the bottom which point upwards towards the open top of the tank. One is a nozzle the other a diffusor. According to the hydrostatic pradox, they shoot the water out to the same height as the water level in the big tank. How come? Why doesn't the law of continuity apply in which the velocity increases at the exit of the nozzle due to the smaller cross section? I though the water out of the nozzle is faster and shoots up higher than the pipe with the wider diameter. I don't understand if and to what extent the hydrostatic paradox aligns with the law of continuity here. According to my textbook, the velocity out of the pipes is the same as the weight force of the water due to gravity.

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Regardless of if a nozzle or a diffuser were to be attached, according to Torricelli's Equation, the velocity of the liquid particles which are just released from a hydrostatic position will be $v = \sqrt{2gh}$ where $h$ is the height difference between the height of the liquid particle and that of fluid in the tank.

Torricelli's Equation

Here in this pic, I have used energy conservation where $dm$ mass is the mass of the sheet of liquid at the surface of nozzle/diffuser and $x$ is the height above the surface which the liquid will rise.

Since $x = h$, the liquid present at the surface of the nozzle/diffuser will shoot out to the same height as the initial height of liquid in the tank.

Things to note are that here, viscosity, air drag and any other dissipative forces are being ignored and if performed in real life, the height gain would be way less than the initial height of liquid in the tank. But the height gained by the liquid should be almost the same in case of both the nozzle and the diffuser

You didn't really provide any information about your textbook so that I could refer to it for what you meant by "the velocity out of the pipes is the same as the weight force of the water due to gravity" but I hope my explanation helped sort out your doubt.

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  • $\begingroup$ Torricelli!! Thank you! $\endgroup$
    – mino
    Jan 27 at 12:03

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