Torricelli's law - Speed of water coming out of an orifice

Problem statement:

The vessel shown in the figure below is filled with water to a height H = 8 m (that we will assume constant). If water is coming out of an orifice at a height h = 3 m from the ground, calculate:

  1. the speed of water coming out of the orifice.
  2. If the orifice were higher up, this speed would be larger or smaller?
  3. By how much should be increased the pressure on the free surface of the water to multiply the speed by three?

Givens: ρ = 103 kg/m3; p0 = 105 Pa; g = 10 m/s2

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Solution:

We are going to assume steady flow. In this conditions we can use Torricelli’s to solve the problem.

Since the height H of water is constant, its speed at the free Surface will be zero. Therefore, applying Torricelli’s law, the speed of water coming out of the orifice is_

Because the height inside the square root is measured from the free surface of the fluid.


It can be seen from the equation above that the closer the orifice will we to the free surface, the lesser the speed will be.


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We are going to use Bernoulli’s equation between states 1 and 2 to solve this part of the problem.

In state (1) the speed of water is zero, and the pressure is the atmospheric pressure plus the increase we want to determine. In (2), pressure is the atmospheric pressure, and the speed of the fluid coming out of the orifice is three times the speed we determined before.

And substituting into Bernoulli’s equation:

Grouping terms and isolating:

Make sure that you include units in your results.

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