Proofs of Torricelli's theorem: Application exercise

Torricelli's Theorem, also known as the Escape Velocity Theorem, is a fundamental principle in fluid mechanics. It is named after the Italian scientist Evangelista Torricelli.
Theorem Statement
The theorem states that the escape velocity of a fluid through an orifice in a container is equal to the velocity the fluid would have if it fell freely from the surface of the fluid to the level of the orifice.
mathematical formula
v = √(2gh)

where:

1. v is the escape velocity of the fluid
2. g is the acceleration of gravity (approximately 9.81 m/s²)
3. h is the height of the fluid above the orifice
Demonstration
The proof is based on the conservation of energy:

1. *Potential energy on the surface*: mgh
2. *Kinetic energy in the hole*: (1/2)mv²

Equalizing both energies:

mgh = (1/2)mv²

Solving for v:

v = √(2gh)
Applications
1. Design of irrigation systems
2. Calculation of flow in pipes
3. Study of erosion
4. Hydraulic engineering
5. Fluid physics
Examples
1. A hole in a container of water 2 meters high has an escape velocity of √(2_9.81_2) ≈ 4.43 m/s.
2. A faucet with a diameter of 1 cm and a height of 5 meters has an escape velocity of √(2_9.81_5) ≈ 9.9 m/s.