Basic Flow Experiment

One of the most important things we use mathematics for is to model real-world situations.  This experiment, which can be run simply and quickly using a 2- or 3-liter soft drink bottle and a rule or tape measure, illustrates the modeling process used throughout the course. 

• This process is essentially the same as that used by scientists, economists, social scientists, business analysts, and anyone who models situations involving functions and changing rates. 
• The behavior of functions with changing rates is the subject of calculus; this course is designed to prepare you for calculus by studying various functions and their behavior.

This experiment involves a physical situation, but we are not doing physics here. 

• Physics involves the reasons the system behaves as it does, due to things like energy conservation, fluid properties, etc. 
• The mathematics of the situation involves linear and quadratic functions, the quadratic equation and systems of simultaneous equations. 
• We aren't going to talk about physics, we aren't going to do physics, we are going to do the mathematics.
• The physical situation is used because it is simple, it gives us precise results, it can be done quickly, and it is very tangible and concrete.

The picture below shows a uniform cylinder filled with water, and set upon a box about a foot high (you will use a soft drink bottle).

• A ruler is taped to the tube (you can use a ruler, a tape measure, or marks made at carefully measured intervals on the tube).
• The idea is to be able to measure the depth of the water above the hole at regular time intervals as it flows out of the hole near the bottom of the tube.
• The instructor is holding his thumb over the hole.

In the next sequence of pictures the water is flowing out of the hole.

The water stream is just barely visible. 

• The pictures are taken at fairly but not precisely regular time intervals.

From the pictures, answer the following:

• Does the depth seem to be changing at a regular rate, at a faster and faster rate, or at a slower and slower rate?
• What do you think a graph of depth vs. time would look like?
• Does the horizontal distance (the distance to the right, ignoring the up and down distance) traveled by the stream increase or decrease as time goes on?
• Does this distance change at an increasing, decreasing or steady rate?
• What do you think a graph of this horizontal distance vs. time would look like?

It is recommended that you perform this experiment, as described below. 

• You will learn a lot more about how mathematics models real-world situations if you take a few minutes to experience the situation; the exercise will be well worth your time. 
• If you prefer, you may get a set of simulated data at the link Simulated Data for Introductory Flow Model

You can easily perform this experiment in a few minutes using a 2- or 3- liter soft drink bottle.  Whether or not you actually perform the experiment or choose to use simulated data, you should read the instructions and understand what is done in the experiment.   If you use simulated data, reading and understanding the instructions will show you what the data means.

• Poke a hole about 1/4" in diameter about an in the side of the bottle about an inch above the bottom.  
• You want the size of the hole to be such that a full bottle will empty through the hole in a minute or two.   A smaller hole and a longer time give better results, but don't go to extremes.
• Fill the bottle about 3/4 of the way full.  
• The bottle, between the filling point and the hole, isn't a perfectly uniform cylinder, but it's reasonably close.  
• Use a ruler or tape measure to measure distance.
• Use a watch to measure the time.
• We will use 'clock time' to refer to the time since the very first reading. The very first reading will therefore be at clock time 0.

You can take data in one of two ways.

• You can write down the clock time every time the water crosses another centimeter line on the ruler.
• Or you can write down the reading on the ruler at regular time intervals, e.g., every 10 seconds.
• You will want about 10 readings, spread out over the time required for the bottle to empty.

You will obtain data which can be put into the following format:

clock time (in seconds, measured from first reading)	Depth of water (in centimeters, measured from the hole)
0	14
10	10
20	7
etc.	etc.

Your numbers will of course differ from those on the table.

Get a ruler, a watch, set the experiment up, and take a careful set of measurements.

The following questions were posed above.  Do your data support or contradict the answers you gave above?

• Is the depth changing at a regular rate, at a faster and faster rate, or at a slower and slower rate?

• What does the graph of depth vs. clock time look like?

   

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Follow the directions given in Overview and Introduction and Completion of the Introductory Flow Model to analyze these data.