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.


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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