09-Mar-2015: Modeling air resistance using coffee filters and logger pro video analysis.

     The purpose of this experiment was to find a relationship between the force of air-resistance and speed. We know that Newtons 2nd Law states that for every reaction there is an equal and opposite reaction. This applies to an object at free-fall under the influence of the force known as weight, calculated by mass times gravity (mg). The force opposing weight is defined as drag, or air resistance. A model of Fairresistance=kv^n was used to determine the relationship between air resistance and speed, where k takes into account the shape and area of the object and n takes into account the material the object is moving through. 

   

     Coffee filters were used as the objects in free-fall. The idea was to drop coffee filters off of a balcony and record the fall. 

Balcony used to drop coffee filters off of. 

     The recorded fall was analyzed by Logger Pro software with the help of human input. Logger Pro allowed people to plot a dot at the location of the coffee filter at any point in time. A position vs time graph comes up. Setting the origin and direction of both axis' gives a graph to each person's liking. Shown here is the analyzed video with the positive y-axis pointing down (the yellow lines). The green line was the reference length measured at 1.46m input into Logger Pro to get accurate data. 

Analyzed recording of the free-fall of 2 coffee filters.

Best fit line showing terminal velocity of 2 coffee filters to be 1.323 m/s.

     Analyzing a best fit line of the resultant position vs time graph (shown above) gave back a terminal velocity of the coffee filters. The terminal velocity being the largest magnitude of velocity when the force of air resistance equals the force of weight. This entire process was repeated 5 times, for 1, 2, 3, 4, and 5 coffee filters falling from the balcony. After those trials, there was enough data to create a force vs velocity graph. Below is the data set input to Logger Pro. In the left column are the terminal velocities of the coffee filters. In the right column is the force of weight acting on the coffee filters, calculated by simply multiplying the mass of 1, 2, 3, 4, or 5 coffee filters by gravity.

     The resultant graph was analyzed with a best fit line. A "power fit" was done since the best fit line needed to correspond with the mathematical model introduced earlier and the points appear to curve up. 

Power fit analyzation provided values for constants k and n.

This power fit produced numbers for k and n that allowed for calculations of terminal velocity. 

k=0.01105

n=1.725

Now that there are numbers for the mass of a coffee filter, k, and n, an excel spreadsheet was opened to help model and predict future experiments. 

There were 6 columns set up in the spreadsheet. They were composed of variables such as time and velocity and contained initial parameters in the free-fall. It looked like:

     Above the variables were reference cells for time, mass, k, and n. This allowed for the change of those variables without having to redo any formulas. This looked like:

     A formula for acceleration was needed to plot into the acceleration column. Doing a simple free-body diagram for the force of air-resistance opposing weight gives an equation for net force of:

Rewriting force as mass times acceleration and solving for acceleration gives:

The spreadsheet was then modeled to make velocity and position as functions of time. Numbers were plugged in and the resulting data was produced. 

Data for mass of 2 coffee filters.

     At first glance, the data might not say much, but after scrolling down the data shows a couple of interesting and useful things.

Data for mass of 2 coffee filters continued..

    First, notice that as time goes on, acceleration approaches zero. Second, notice that as acceleration approaches zero , velocity rises and levels off. 

     Acceleration approaching zero means that the force of air-resistance is becoming equal to the force of weight. And if those two forces are becoming equal to each other, then the object must be approaching terminal velocity. So the excel model works.

     

     How did it compare to the experiment data? 

The excel model compares well with the experiment data. The largest difference in terminal velocities was for one coffee filter.

Experiment terminal velocity=0.9578 m/s

Excel model terminal velocity=0.8745 m/s

Difference in velocities=0.0833 m/s

The smallest difference in terminal velocities was for 2 coffee filters, a difference of 0.016 m/s.

So, the excel model has a close enough approximation to be able to predict how experiments will turn out.