Torricelli's Law (v = √(2gh))

Water leaving a hole in the side of a tank comes out exactly as fast as if it had fallen freely from the surface down to the hole. The argument is energy conservation along a streamline: a parcel starts at the free surface, where the pressure is atmospheric and it is barely moving, and arrives at the opening, where the pressure is atmospheric again and it is moving fast. The pressure terms are equal and cancel, leaving $mgh = \tfrac{1}{2}mv^2$, and the mass cancels too. What survives is $v = \sqrt{2gh}$ — the free-fall speed.

A tap 5 m below the water line of a standpipe jets out at $\sqrt{2\times9.80665\times5} = 9.90$ m/s. Through a 25 mm opening (area $4.91\times10^{-4}$ m²) that would be $Q = 4.86\times10^{-3}$ m³/s, or 292 L/min — if the opening passed the full theoretical flow, which it does not. Hold that number; the last paragraph collects on it.

Evangelista Torricelli published this in 1643. He was Galileo's secretary in the old man's final months and inherited both his notes and his instinct for stripping a problem to its mechanics, and it is the same Torricelli who invented the barometer a year earlier. The two results belong together: one says a column of fluid produces a pressure, the other says that pressure spends itself as speed. Notice what is absent from the formula — the fluid's density. Mercury and water pour from the same depth at the same speed, for the same reason a bowling ball and a marble fall alike. Read alongside the pressure-head and velocity-head pages, this law is simply the statement that one head converts entirely into the other.

The real discharge is far below the ideal, and this is where the equation misleads people. Fluid approaching a sharp-edged hole arrives from all directions and cannot turn the corner instantly, so the jet keeps contracting after it leaves — the vena contracta — narrowing to about 62% of the hole area a short distance out. Add a small velocity loss and the discharge coefficient for a sharp-edged orifice lands near 0.61. That 292 L/min is really about 180. Round the entry into a bellmouth and $C_d$ rises to roughly 0.98, which is why a well-formed nozzle passes half again as much water as a drilled plate of the same diameter.

Three more conditions. $h$ is measured from the free surface down to the opening, not from the top of the tank and not from the bottom — and it shrinks as the tank drains, so this gives the speed at one instant, not an average. Working out how long a tank takes to empty means integrating, which produces the pleasant result that draining time scales with the square root of the starting depth. The tank must also be vented: seal it and the flow chokes as a partial vacuum forms above the water, which is why a full jerry can glugs instead of pouring. And if the space above the liquid is deliberately pressurised, that pressure adds its own head on top of $h$, and the simple form no longer applies on its own.