1. Objective

The purpose of this lab is to experimentally prove and demonstrate the fluidal relationship between velocity v and height h.

2. Proof of Torricelli’s Equation

Previously, our AP Physics II class has only come into contact with the relation between fluidal velocity and fluidal height through the abstract formula that constitutes Bernoulli’s Equation:

\[P + \rho \textit{gh} + \frac {\rho\textit v^2}{2} = constant\]

, where \(P\) is fluidal pressure, \(\rho\) is density of the liquid, \(g\) is gravity, otherwise known as approximately \(9.8 m/s^2\), \(h\) is height, \(\textit v\) is the velocity at which the fluid moves, and \(constant\) is the constant, total energy that the system contains. Bernoulli’s Equation is derived from the fact that in fluids, there are three kinds of energy: fluidic (\(P\)), gravitational (\(\rho \textit{gh}\)), and kinetic (\(\frac {\rho\textit v^2}{2}\)) and that if we have a closed system, then it is safe to say that there will be no energy loss or gain, as in the Conservation of Energy. Additionally, since the energy will be constant throughout the system, we may establish two points within the system and set their respective equations equal to one another.

\[P_1 + \rho \textit gh_1 + \frac {\rho\textit {v}_1^2}{2} = constant\]

\[P_2 + \rho \textit gh_2 + \frac {\rho\textit {v}_2^2}{2} = constant\]

Recognize that the \(\rho\) and the \(g\) are equivalent at both positions 1 and 2, as the density of the liquid has not changed (given that these fluids are incompressible) and that \(g\) is a gravitational force that is realized at all points in the system.

\[P_2 + \rho \textit gh_2 + \frac {\rho\textit {v}_2^2}{2} = P_1 + \rho \textit gh_1 + \frac {\rho\textit {v}_1^2}{2}\]

Now, it is possible to combine the two equations, and simplify.

\[P_2 - P_1 = \rho \textit gh_1 - \rho \textit gh_2 + \frac {\rho\textit {v}_1^2}{2} - \frac {\rho\textit {v}_2^2}{2}\]

\[\Delta P = P_2 - P_1 = \rho \textit gh_1 - \rho \textit gh_2 + \frac {\rho\textit {v}_1^2}{2} - \frac {\rho\textit {v}_2^2}{2}\]

\[\Delta P = P_2 - P_1 = \rho \textit g(h_1 - h_2) + \frac {\rho (\textit {v}_1^2 - {v}_2^2)}{2}\]