Torricelli's law
Torricelli's law, also known as Torricelli's theorem, is a theorem in fluid dynamics relating the speed of fluid flowing from an orifice to the height of fluid above the opening. The law states that the speed "v" of efflux of a fluid through a sharp-edged hole at the bottom of the tank filled to a depth "h" is the same as the speed that a body (in this case a drop of water) would acquire in falling freely from a height "h", i.e. formula_1, where "g" is the acceleration due to gravity. This expression  comes from equating the kinetic energy gained, formula_2, with the potential energy lost, "mgh", and solving for "v". The law was discovered (though not in this form) by the Italian scientist Evangelista Torricelli, in 1643. It was later shown to be a particular case of Bernoulli's principle. Derivation. Under the assumptions of an incompressible fluid with negligible viscosity, Bernoulli's principle states that the hydraulic energy is constant Since the water level is formula_4 above the orifice, the horizontal efflux velocity formula_5 as given by Torricelli's law. Thus, we have from the two equations The location of the  orifice that yields the maximum horizontal range is obtained by differentiating the above equation for formula_7 with respect to formula_8, and solving formula_9. Here we have Solving formula_11 we obtain and the maximum range Clepsydra problem. A clepsydra is a clock that measures time by the flow of water. It consists of a pot with a small hole at the bottom through which the water can escape. The amount of escaping water gives the measure of time. As given by the Torricelli's law, the rate of efflux through the hole depends on the height of the water; and as  the water level diminishes, the discharge is not uniform. A simple solution is to keep the height of the water constant. This can be attained by letting a constant stream of water flow into the vessel, the overflow of which is allowed to escape from the top, from another hole. Thus having a constant height, the discharging water from the bottom can be collected in another cylindrical vessel with uniform graduation to measure time. This is an inflow clepsydra. Alternatively, by carefully selecting the shape of the vessel, the water level in the vessel can be made to decrease at  constant rate. By measuring the level of water remaining in the vessel, the time can be measured with uniform graduation. This is an example of outflow clepsydra. Since the water outflow rate is higher when the water level is higher (due to more pressure), the fluid's volume should be more than a simple cylinder when the water level is high. That is, the radius should be larger when the water level is higher. Let the radius formula_14 increase with the height of the water level formula_8 above the exit hole of area formula_16 That is, formula_17. We want to  find the radius such that the water level has a constant rate of decrease, i.e. formula_18. At a given water level formula_8, the water surface area is formula_20. The instantaneous rate of change in water volume is From Torricelli's law, the rate of outflow is From these two equations, Thus, the radius of the container should change in proportion to the quartic root of its height, formula_24 Likewise, if the shape of the vessel of the outflow clepsydra cannot be modified according to the above specification, then we need to use non-uniform graduation to measure time. The  emptying time formula above tells us the time should be calibrated as the square root of the discharged water height, formula_25 More precisely, where formula_27 is the time taken by the water level to fall from the height of formula_28 to height of formula_29. Torricelli's original derivation. Evangelista Torricelli's original derivation can be found in the second book 'De motu aquarum' of his 'Opera Geometrica' (see ): He starts a tube AB (Figure (a)) filled up with water to the level A. Then a narrow opening is drilled at the level of B and connected to a second vertical  tube BC. Due to the hydrostatic principle of communicating vessels the water lifts up to the same filling level AC in both tubes (Figure (b)). When finally the tube BC is removed (Figure (c)) the water should again lift up to this height, which is named AD in Figure (c). The reason for that behavior is the fact that a droplet's falling velocity from a height A to B is equal to the initial velocity that is needed to lift up a droplet from B to A. When performing such an experiment only the height C (instead of D  in figure (c)) will be reached which contradicts the proposed theory. Torricelli attributes this defect to the air resistance and to the fact that the descending drops collide with ascending drops. Torricelli's argumentation is, as a matter of fact, wrong because the pressure in free jet is the surrounding atmospheric pressure, while the pressure in a communicating vessel is the hydrostatic pressure. At that time the concept of pressure was unknown.