Visualizing properties of a quadratic function using Torricelli’S fountain

Abstract

In the same chapter of his book Opera geometrica, Torricelli published1 two discoveries: 1) initial velocity of a jet from a container increases with the square root of the depth of the hole¸ 2) he draw the pattern of jets from three openings at the wall of a container filled with water to constant level H and determined the height of the hole with maximal range. In studying the pattern Torricelli used the mentioned law of initial velocities and Galileo’s law of free fall and projectile motion. The first Torricelli’s discovery is now well known in physics education under the name Torricelli’s law. But the pattern of jets from a container entered into physics literature along two ways, which we propose to name: “da Vinci’s way” and “Torricelli’s way”. Along “da Vinci’s way” educators and textbook authors (Ref. 2 and textbooks and articles cited by Biser3 and Atkin4) present incorrect drawings of jets in order to incorrectly “demonstrate”the correct Torricelli’s law. Along “Torricelli’s way” educators point out3-11 that the shape and range of a jet depend on the initial velocity as well as on the time of flight of a jet. Using algebra and calculus (instead of geometry, proportions and narrative used by Torricelli and Galileo) the shape of trajectories, their envelope, range and meeting of two jets at an arbitrary datum level, are determined by quadratic function and quadratic equation. Their detailed mathematical analysis is presented in this paper. In describing how the use of water and air through time has developed our scientific understanding, and how to bring fluid mechanics to the general public, E. Guyon and M. Guyon12 observed: “Water fountains and jets are still being built and are favorite public attractions but, alas, are seldom connected to their scientific meaning, unlike the Torricelli fountain shown in Fig. 1.”