# Torricelli’s Law

## Torricelli’s Law

Torricelli’s law, also known as Torricelli’s theorem, statement in fluid dynamics that the speed (v), of fluid flowing out of an orificeunder the force of gravity in a tank is proportional to the square root of the vertical distance between the liquid surface and the centre of the orifice and to the square root of twice the acceleration due to gravity.

The efflux velocity of the fluid from the orifice is the same as that it would have acquired by falling a height under gravity. The law was discovered by and named after the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli’s principle.

Consider any vessel which has an orifice filled with some fluid. The fluid will start flowing through the slit and according to Torricelli law the speed with which the fluid will flow is equal to the speed with which a freely falling body attains such that the height from which the body falls is equal to the height of the slit from the free surface of the fluid.

Torricelli’s Law derivation:

Torricelli’s Law is derived from the Bernoulli equation by equating the potential energy in the static head of liquid to the kinetic energy of the water leaving through the orifice as follows:

$$\frac{{{P}_{1}}}{\rho g}+\frac{v_{1}^{2}}{2g}+{{z}_{1}}=\frac{{{P}_{2}}}{\rho g}+\frac{v_{2}^{2}}{2g}+{{z}_{2}}$$,

$$\frac{0}{\rho g}+\frac{0}{2g}+{{z}_{1}}=\frac{0}{\rho g}+\frac{v_{2}^{2}}{2g}+0$$,

$$z_{1}^{2}=\frac{v_{2}^{2}}{2g}$$,

Subsequently rearranging for velocity, we have Torricelli’s law:

$${{v}_{2}}=\sqrt{2g{{z}_{1}}}$$,

A simple experiment to test Torricelli’s law can also be performed by a soda bottle by puncturing the bottom with a small hole. As the height in the reservoir decreases, the exit velocity decreases.