Visualizing Properties of a Quadratic Function Using Torricelli’s Fountain

Abstract

In the same chapter of his book Opera Geometrica, Torricelli published two discoveries: 1) initial velocity of a jet from a container increases with the square root of the depth of the hole; 2) he drew 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 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 Biser and Atkin) present incorrect drawings of jets in order to incorrectly “demonstrate” the correct Torricelli’s law. Along “Torricelli’s way” educators point out 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.