Prediction and Controls
Click the shapes in the order you think they will empty, from fastest to slowest.
Derivation and Interpretation
An object dropped from height \(h\) acted on only by gravity hits the ground with velocity \(v=\sqrt{2gh}\). Torricelli's observation was that water draining from a hole a distance \(h\) below the water level also exits with this same speed. Using the hole area \(A_h\), this gives the outflow law
More generally, if \(A(h)\) is the cross-sectional area of the container at height \(h\), then \(\frac{dV}{dh}=A(h)\), so related rates gives the general differential equation
or equivalently
For a cylinder, for example, \(V=\pi R^2h\), so \(\frac{dV}{dt}=\pi R^2\frac{dh}{dt}\). The equation is harder to solve for cones, spheres, and paraboloids because the cross-sections are not constant. The visual above uses the solutions for all five cases, and for the sphere some numerical approximation is used.
Notice how objects with larger cross-sections higher in the tank drain faster because the velocity of the liquid coming out is higher longer.
Where It Started
Aiden asked to look at a Math 124 draining-from-a-cone question and made a visual from it. That led to a broader comparison of how different shapes drain when they begin with the same volume and have the same hole size.
Dr. Loveless then explained Torricelli's law, which he first encountered while teaching Math 207 many years ago, and later turned into a separate project of his own.
Going Further
This same model can be used to study how leaks at different heights affect how far the stream of water comes out from the side of a container.
There is a lot of interesting physics and math here, including projectile motion, differential equations, geometry, and how well idealized models match experiment.
Links
Torricelli’s Law and How Long to Drain a Bathtub
This project handout gives more background, a derivation, and several follow-up questions.