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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

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