**Bernoulli's principle** (or **Bernoulli's equation**) is a formula that relates the height, density, pressure, and velocity of a non-viscous and non-conducting fluid. It states that

- $ p\, +\, \rho\, g\, h\, +\, \tfrac12\, \rho\, v^2\,=\, \text{constant}\, $

*p* is the pressure, *ρ* is the density, *h* is the elevation, *v* is the velocity of the fluid, and *g* is acceleration due to gravity.

## DerivationEdit

Since total energy in a system is conserved,

- $ W + E_p + E_k = \text{constant} $

Work is equal to force times distance. Since pressure is force divided by area and distance is velocity times time, we get

- $ W = Fd = \frac{F}{a} a\, v\, t\, = p\, a\, v\, t\, $

Since area times velocity times time is equal to volume, we can rewrite this as mass over density, giving

- $ W = \frac{p\, m\,}{\rho} $

The potential energy will be equal to

- $ m\, g\, h\, $

and the kinetic energy will equal

- $ \tfrac12\ m\, v\, ^2 $

By substituting these values in, we get

- $ \frac{p\, m\,}{\rho} + m\, g\, h\, + \tfrac12\ m\, v\, ^2 = \text{constant} $

This can be simplified by dividing both sides by *m* and multiplying by *ρ*, giving

- $ p\, + \rho\, g\, h\, + \tfrac12\ \rho\, v\, ^2 = \text{constant} $