calculus 2 assignment

Math 266: Calculus for Engineers II

Project Four

Down the Drain…

The Project will address modelling of physical phenomena through the use of Differential Equations, and

the refinement of models through experimentation.

Note: Unlike previous Projects, this Project will requires the collection of data through experimentation.

Be sure to budget your time accordingly. You will also need a two-liter soda bottle that is more or less

cylindrical. In other words, not one from a brand that has trademarked a particular curved shape.

Gravity has been used to supply water for all of human history. In 1643 Evangilista Torricelli formalized

our understanding of the relationship between water and gravity looking at the potential energy of a

particle of water at a given height. Today, (one form of) Torricelli’s Law tells us:

𝑑𝑉

𝑑𝑡 = −𝑎√2𝑔ℎ,

Where V is the Volume of liquid, a is the area of the hole through which it is passing, g is the force due to

gravity, and h is the height of the liquid. Show that for a cylindrical container of radius R with a circular

outflow hole of radius r, Torricelli’s Law reduces to:

𝑑ℎ

𝑑𝑡 = − (

𝑟

𝑅 )

2

√2𝑔ℎ

Assuming that you drill a hole 4mm in diameter in your liter bottle, solve the above Differential Equation

for h(t), given h0 = 10 cm. According to this solution, how long will it take for water level to drop to a

height of 3cm?

Torricelli’s Law is a mathematical ideal. Because of viscosity, surface tension, and how the liquid

interacts with the shape of the outflow hole, actual flow rates will be slightly different. It turns out that

all of these differences can be summarized in a single coefficient, giving:

𝑑ℎ

𝑑𝑡 = −𝑘√ℎ

Drill an actual hole 4 mm in diameter in the side of your actual two-liter bottle, near the bottom but still

in the cylindrical zone. Fill the bottle to a height 10 cm above the hole, and time how long it takes to

drop to 3 cm above the hole. How does this time compare to the ideal solution? Using this information,

solve for the value of k. How does the ideal solution for h(t) compare to the one found using your

experimental model?