# Torricelli's Law

Torricelli's Law, named after Evangelista Torricelli, relates initial velocity, gravity, and height to the flow of a liquid out of an opening: $$v_2=\sqrt{2gh+v_1^2}$$ where $v_2$ is the speed out of the opening, $v_1$ the initial speed, $g$ gravity, and $h$ the distance from the top of the hole to the top of the liquid.

## Derivation

Torricelli's Law may be derived by considering the mass of water $m$ at the top of the liquid. This water has potential energy $mgh$ relative to the hole and kinetic energy $\frac 12 mv_1^2$, for total energy of $m(gh+\frac 12 v_1^2)$. At the level of the hole, this water will have converted all potential energy into kinetic energy, so the total energy will now be $\frac 12 mv_2^2$. By the First Law of Thermodynamics, $T_1=T_2$, so $m(gh+\frac 12 v_1^2)=\frac 12 mv_2^2$; solving for $v_2$, $$v_2=\sqrt{2gh+v_1^2}$$