Bernoulli's Equation: Formula, Examples & Problems

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  • 0:01 Fluid Motion Is Conserved
  • 1:57 Bernoulli's Equation
  • 2:53 Examples of…
  • 6:00 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

Bernoulli's equation describes an important relationship between pressure, speed, and height of an ideal fluid. In this lesson you will learn Bernoulli's equation, as well as see through an example how in an ideal fluid, the dynamics of that fluid remain constant.

Fluid Motion Is Conserved

If you're like me, you really dig recycling. It's amazing that we can take a material that has been used for one purpose and then either change its form or function so that it can be used again for another purpose. The universe likes recycling, too!

In fact, all of the energy on Earth, other planets, and the rest of the universe is constant - it is not created or destroyed, only transferred from one object to another, or transformed from one form to another. This is concept is so vital to how things work in the universe that it's called the law of conservation of energy.

We can apply the same idea of conservation to fluids. This is because as a fluid moves through a pipe or tube, the relationship between the pressure, velocity, and height remains constant. When one of these variables changes, that change is 'recycled' as a respective change in another variable.

For example, if the pressure in the fluid increases, the speed of the fluid decreases to compensate. Likewise, if the area that the fluid travels through becomes smaller, the speed increases because the same amount of volume has to travel through that smaller area.

However, there are some caveats attached to these fluids that make them 'ideal.' Fluid dynamics is a very complex subject, and we don't even fully understand some of the ways that fluids move. So we have to make a few assumptions to create an 'ideal' fluid that allows us to understand its movement.

First we assume that the fluid is incompressible, meaning that its density doesn't change. Second, we assume that the fluid is non-viscous, meaning that there is no resistance to the fluid's movement. Finally, we assume that the flow is laminar, which means that it is steady and constant. It would be quite difficult if we tried to work with a fluid that was moving around all willy-nilly on us.

Bernoulli's Equation

We can neatly package the concept of fluid conservation in Bernoulli's equation, which relates pressure, speed, and height at any two points within an ideal fluid.

This relationship can be written as an equation:

P1 + ½ ρv1^2 + ρgh1 = P2 + ½ ρv2^2 + ρgh2

where P is the pressure in the fluid, ρ is the density of the fluid, g is the acceleration due to gravity (9.80 m/s^2), h is the height of the fluid off the ground, and v is the velocity of the fluid.

Can you see how if one variable changes at point 1, then something else must also change to keep the equation, well, equal?

Example of Bernoulli's Equation

You may still be having some difficulty grasping this concept and relating it to the conservation of energy, so let's work through an actual example.


Say that some water flows through an S-shaped pipe. At one end, the water in the pipe has a pressure of 150,000 Pascal (Pa), a speed of 5.0 m/s, and a height of 0.0 m. At the other end, the speed of the water is 10 m/s, and the height is now 2.0 m. Since the density of water is 1000 kg/m^3, all you're missing is the pressure at the second point, and this can be determined by rearranging Bernoulli's equation to get it alone on one side.

So to start, our equation looks like this:

P1 + ½ ρv1^2 + ρgh1 = P2 + ½ ρv2^2 + ρgh2

To make things easier, let's rearrange our equation first, and then plug in our values. To get P2 alone, we rearrange things so our equation looks like this:

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