10-Sept-2014: Modeling the fall of an object with air resistance

The purpose of this lab is to determine the relationship between air resistance force and speed.

The apparatus used was numerous coffee filters that were the object being measured for speed and air resistance, and a meter stick that determined the distance the filter was being dropped

To begin the experiment we discussed what forces were being applied on the coffee filter(s) as they journeyed downward. We constructed a free body diagram of the coffee filter(s) as they were in free fall and determined the forces, The forces acting on the coffee filter(s) were its weight (mass x gravity) and an unknown air resistance force. We were given the equation for air resistance which was:

Resistance force = k x (velocity^n)

Knowing the two forces we were now able to find an a equation for sum of the forces whcich allowed us to find the acceleration of the system.

ΣF = mg - k(v^n)

We want to know the relationship between speed and air resistance so by dividing the mass from the left side we are left with the value of acceleration.

a = g - (k/m)(v^n)

We are able to find the values of all the variables except for k and n which are constant numbers. In order to find their values we must conduct the experiment. 

Using a laptop's webcam we record the descent of the filters beginning with one filter, and adding one each trail. We record its descent of the first meter it travels downward. Using logger pro we are able to analyze the video at every tenth of a second and determine the position of the filter thus giving us the velocity of the filter at a specific time. At some point the filter(s) stop accelerating and move at a constant velocity or terminal velocity. At this point the weight of the filter(s) equal the air resistance causing the acceleration of the systems to equal zero. By constructing graphs for each trail we are able to determine what the terminal velocity of each one is. These are the five graphs where the slope represents the terminal velocity

1 Filter

2 Filters

3 Filters

4 Filters

5 Filters

Now that we know the value of terminal velocity we are able to substitute it in the equation for air resistance. Since the acceleration is equal to zero when the filter(s) reach terminal velocity the force of air resistance must equal the weight of the filter(s). By measuring the mass of one filter we are able to calculate the mass used for each trail. The mass of one filter is 0.932 grams or 9.32 x 10^-4 kilograms. With the results we compare the terminal velocity vs. the air resistance experienced by the filters and graph it.

Using a power fit we are given the range of the values for k and n as seen above where A represents k and B represents n. k = (-0.00176, 0.007141) and n = (-0.2359,1.651)

The second part of the lab required us to model the fall of the filters including air resistance using excel. Here we chose two trails to model those being the 1 filter and 5 filters. The spreadsheet was set up to contain time in intervals of 0.002 seconds, acceleration, velocity relative to time, and terminal velocity. The results below show the first 10 columns of the data and the time interval where the filter(s) reach terminal velocity

Filter 1

 Filter 2

In Conclusion it is clearly depicted that the terminal velocity is similar in both logger pro and excel. We saw the relationship between the terminal velocity of the object with the air resistance it experiences learning that the terminal velocity is caused by the equal values of the filter(s) weight and the air resistance.