Torricelli's Theorem
Torricelli's theorem v = sqrt(2gh) calculates the speed of fluid flowing from an opening in a container.
Learn with examples.
The Formula
Torricelli's theorem gives the speed of a fluid flowing out of an opening in a container. The exit velocity depends only on the height of the fluid above the opening, not on the size of the hole or the density of the fluid.
Italian physicist Evangelista Torricelli discovered this principle in 1643. It is actually a special case of Bernoulli's equation applied to a fluid draining from a tank. The formula is identical to the velocity of a freely falling object dropped from height h, which is a remarkable and elegant result.
The theorem assumes the fluid is ideal (no viscosity), the tank is large compared to the opening (so the surface drops slowly), and the opening is exposed to the same atmospheric pressure as the surface. For real fluids, the actual flow rate is typically 60-65% of the theoretical value due to the contraction of the fluid stream at the opening.
Variables
| Symbol | Meaning |
|---|---|
| v | Velocity of fluid at the opening (m/s) |
| g | Acceleration due to gravity (9.81 m/s²) |
| h | Height of fluid surface above the opening (meters, m) |
Example 1
A water tank has a small hole 3 meters below the water surface. What is the exit velocity?
Apply the formula: v = √(2gh) = √(2 × 9.81 × 3)
v = √(58.86)
v ≈ 7.67 m/s
Example 2
Water exits a tank at 5 m/s. How high is the water level above the outlet?
Rearrange: h = v²/(2g) = 5²/(2 × 9.81)
h = 25/19.62
h ≈ 1.27 m above the outlet
When to Use It
Use Torricelli's theorem to calculate the speed of fluid draining from a container through an opening.
- Designing drainage systems and water tanks
- Estimating flow rates from reservoirs and dams
- Calculating how quickly a tank will empty
- Engineering irrigation and plumbing systems