Torricelli's Law Calculator

At a glance:Torricelli's law (Evangelista Torricelli, 1643, Italian physicist, pupil of Galileo) describes the speed of a liquid jetting from a small hole in the side or bottom of a container: it depends only on the head h (the vertical height of water above the hole), not on the liquid's type or density. The formula is strikingly simple: v=√(2·g·h), with v the exit speed (m/s), g gravitational acceleration (9.81 m/s²) and h the head (m). It comes from energy conservation (Bernoulli's equation): comparing the mechanical energy at the free surface and at the orifice, the surface potential energy ρgh converts into kinetic energy ½ρv² at the hole; cancelling density ρ gives v²=2gh, and taking the square root yields the formula. A very intuitive meaning: the jet speed from the hole equals exactly the speed a water drop would reach after free-falling the height h — so orifice outflow and free fall share the same physics, the most charming part of Torricelli's law. Term by term: h is the effective head, the vertical distance from the water surface to the orifice centre; higher surface means higher pressure and faster outflow. g=9.81 m/s². Important: this v is the ideal (inviscid, lossless) exit speed. A real orifice, because of flow contraction (the vena contracta, the jet necking) and viscous friction, gives smaller actual speed and flow than the ideal — engineering multiplies by a velocity coefficient Cv≈0.97–0.99 and a contraction coefficient Cc, combining into a discharge coefficient Cd (sharp-edged orifice ≈0.6–0.62). So the actual flow is Q=Cd·A·√(2gh), not simply ideal v times area. Example: head h=2 m → v=√(2×9.81×2)=√39.24≈6.264 m/s. If the orifice area A=0.001 m² and Cd=0.62, the real flow is Q=0.62×0.001×6.264≈0.00388 m³/s. Uses: (1) estimate outflow speed/flow from reservoir, water tower and irrigation tank drain holes; (2) size drain and overflow holes; (3) control flow in water clocks and fountains; (4) demonstrate energy conservation and free fall in teaching. Notes: (1) this tool gives the ideal exit speed — multiply by Cd for real flow; (2) the small-orifice assumption needs the orifice much smaller than the head, with an approximately steady surface (large tank or continuous refill); (3) if the level drops while draining, h varies and the emptying time needs integration; (4) h is the vertical distance from surface to orifice centre (m). In short, Torricelli's exit speed is v=√(2gh), equal to the free-fall speed — the basis of orifice outflow and tank-draining estimates.