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 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.
Derivation[]
Since total energy in a system is conserved,
Work is equal to force times distance. Since pressure is force divided by area and distance is velocity times time, we get
Since area times velocity times time is equal to volume, we can rewrite this as mass over density, giving
The potential energy will be equal to
and the kinetic energy will equal
By substituting these values in, we get
This can be simplified by dividing both sides by m and multiplying by ρ, giving