Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from a hole to the height of fluid above the hole. The law states that the speed ${\displaystyle v}$ of efflux of a fluid through a sharp-edged hole in the wall of the tank filled to a height ${\displaystyle h}$ above the hole is the same as the speed that a body would acquire in falling freely from a height ${\displaystyle h}$,

${\displaystyle v={\sqrt {2gh}}}$

where ${\displaystyle g}$ is the acceleration due to gravity. This expression comes from equating the kinetic energy gained, ${\displaystyle {\tfrac {1}{2}}mv^{2}}$, with the potential energy lost, ${\displaystyle mgh}$, and solving for ${\displaystyle v}$. The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle.

Under the assumptions of an incompressible fluid with negligible viscosity, Bernoulli's principle states that the hydraulic energy is uniform

throughout a column of liquid. Here ${\displaystyle v}$ is fluid speed, ${\displaystyle g}$ is the acceleration due to gravity, ${\displaystyle y}$ is the height above some reference point, ${\displaystyle p}$ is the pressure, and ${\displaystyle \rho }$ is the density.

In order to derive Torricelli's formula the first point with no index is taken at the liquid's surface, and the second just outside the opening. Since the liquid is assumed to be incompressible, ${\displaystyle \rho _{1}}$ is equal to ${\displaystyle \rho _{2}}$ and; both can be represented by one symbol ${\displaystyle \rho }$. The pressure ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ are typically both atmospheric pressure, so ${\displaystyle p_{1}=p_{2}\Rightarrow p_{1}-p_{2}=0}$. Furthermore ${\displaystyle y_{1}-y_{2}}$ is equal to the height ${\displaystyle h}$ of the liquid's surface over the opening:

The velocity of the surface ${\displaystyle v_{1}}$ can by related to the outflow velocity ${\displaystyle v_{2}}$ by the continuity equation ${\displaystyle v_{1}A=v_{2}A_{A}}$, where ${\displaystyle A_{A}}$ is the orifice's cross section and ${\displaystyle A}$ is the (cylindrical) vessel's cross section. Renaming ${\displaystyle v_{2}}$ to ${\displaystyle v_{A}}$ (A like Aperture) gives:

Torricelli's law is obtained as a special case when the opening ${\displaystyle A_{A}}$ is very small relative to the horizontal cross-section of the container ${\displaystyle A}$: