1. Modeling the height of a falling column of water.
Collect data and plot data,
Develop a physical based mathematical model,
Solve the mathematical model
Estimate parameters in the model
Iterate and Validate

2. Collect data and plot data
If you have a computer then you can{
Connect to the Network ID/SSID of “Hilton Chicago Meeting”
Enter password "MATHFEST2017“
Type “SIMIODE YouTube” in your Browser
and select a video with column of water and clock and collect data.
Otherwise we will give you data from a run of one of the videos.
We will use “our” data, not yours :+( for the rest of this talk.

6. The sum of the potential energy and the kinetic energy of a particle
We first consider a law of physics which can help us.
The Law of Conservation of Energy says
The sum of the potential energy and the kinetic energy of a particle
of mass m is constant, i.e. total energy is conserved.
We consider a particle of water of mass m initially atop a cylinder of water,
some h meters above a small, sharp-edged opening in the side wall of the cylinder
through which the water can exit the cylinder
This mass of water has initial potential energy PEi = m*g*h, g is the acceleration
due to gravity, and initial kinetic energy KEi = ½*m*vi 2
where vi is the initial velocity of the mass.
Thus, we have an initial total energy of TEi when the mass of water is on the
top of the cylinder of water:
TEi = KEi + PEi = ½*m*vi 2 + m*g*h

7. Assuming the mass of water is not moving at the top, solve for vf.
When this mass of water reaches the opening at the bottom of the column’s
container it has height 0 meters above the opening and a final velocity of vf .
Hence, the total energy at the final time (TEf ) the mass reaches the opening is
TEf = KEf + PEf = ½*m*vf 2 + m*g*0
Thus by the Conservation of Energy Law the Total Energy of the mass at the top, TEi
initially, equals the Total Energy of the mass at the bottom. TEf finally.
½*m*vi 2 + m*g*h = TEi = TEf = ½*m*vf 2 + m*g*0 = ½*m*vf 2
Assuming the mass of water is not moving at the top, solve for vf.

8. How did you do?

9. Did you get the following?
How did you do?
Did you get the following?
Torricelli “did” and so we call it Torricelli’s Law.
Evangelista Torricelli, an Italian physicist
and mathematician(1608 –1647), who
invented the barometer.

10. So the water at the bottom of the column
of water has velocity Vf and the only way
to go is out the small bore hole. So OUT!!

11. Let us now identify variables:
h(t) is the height of water in meters at time t in seconds,
a is the cross sectional area of the bore hole at the bottom of the column, and
A(h(t)) is the cross sectional area of the cylinder forming the column of water.
So knowing Torricelli’s Law above
derive the differential equation for h(t) by equating two ways to calculate
the amount of water leaving the cylinder at time t. What might a represent?

12. A small hole of radius s = 0.138906 cm is drilled in the side
of a right circular cylindrical container of radius r = cm
and the height of the water level (above the hole) goes
from 12.5 cm at a stop watch time of 1: min
to 2 cm at a stop watch time of 2:34.116

16. Question for you to consider.
What if we took a cylinder of water (right)
and placed some blocks into it (left),
thus raising the level BUT NOT THE AMOUNT of
water in the cylinder.
Now open the little bore hole in the bottom of
each at the same time and watched?
Which would empty fastest? Or do they empty
in the same amount of time?
Model it!