Lab 4 - March 22, 2015 - Modeling the Fall of an Object with Air Resistance

Lab 4:
3/22/2015
Brandon Elder

Modeling the Fall of an Object with Air Resistance

Purpose: This lab is two-part, first we will determine the relationship between air resistance and speed, second we will model the fall of an object including air resistance.

Part 1: First of all, we expect that the force of air resistance can be modeled by some exponential equation, such as below (See Fig. 1).

Fig. 1 The force of air resistance is equal to some constant
times the velocity to some degree.

Fig. 3 The filter reaches
terminal velocity and
no longer requires
acceleration to be factored
into the equation.

In order to measure these unknown terms and to determine the air resistance, we will drop coffee filters (see Fig. 2) from a set height and record the speed at which they float down. At some point along this drop down, they will reach their terminal velocity, where there will no longer be any acceleration (see Fig. 3) acting on the filters, so the sum of forces will be the air resistance equal to some equation we will determine.

Fig. 2 We used a total of 5 coffee filters.

We headed over to the technology building, building 13 because there is a good place to drop the filters from inside where we can record them falling down uninterrupted (see Fig. 4). We set up LoggerPro to capture video for each drop. We started with one filter. Then we added another filter to the first one. We were careful to add each additional filter inside the first one, so we could limit the change in the surface area and shape of the filter being recorded as it dropped down. We repeated this process until we had a video for one to five filters, for a total of five drops.

Fig. 4 The white ledge at the bottom of our feet was measured so that the known distance could be inputted into LoggerPro later as a reference mark for distance. The filters were dropped and allowed to float all the way down to the ground in a straight line.

A video was taken of this entire process and each drop was stored in it's own file. Back in the lab, we evaluated each of these videos. Through LoggerPro, we placed a dot on the falling coffee filter every three frames. The software assigns a time value to the location we click on. The computer stores these points in a location and time table. This is done for each video until the filter either reaches the ground or is very near to the ground (see Fig. 5).

Fig. 5 A dot was placed every few frames
capturing the location of the coffee filter vs time.

Fig. 6 Slope of graphs has the
terminal velocity.

The picture below shows the videos we captured during our data collection and the graph shows our data points acquired through the video capture. The slope near the end of the curve signifies the terminal velocity reached by each of the coffee filters (see Fig. 6). We need the terminal velocity in order to graph the Position vs Time. This process was done 5 times for each of the coffee filters used.

Below is the graph of Speed vs. Force that is used to determine k and n in the equation Air Resistance Force = kv^n (see Fig. 7).  The third data point was not included because we thought we would get a better correlation if we excluded it. We got 0.007 for k and 1.793 for n.

Fig. 7 Graph of Speed vs Force used to determine
k and n in the equation F=kv^n

Fig. 8 Data plugged into an excel spreadhseet
 set to calculate the terminal velocity.

Calculations: In order to verify that we were able to get the correct values, we put the values of k and n into excel (see Fig. 8) in order to find the final velocity. The results proved to be the same, so it was accurate. Below are our calculations that we did for the experiments. (See Fig. 9). Fig 10 has the equation that we used to determine the weight of one coffee filter.

Fig. 9 All data calculated
from the experiment.

Fig. 10 Calculation used to determine
the weight of one coffee filter.

Summary: First, we dropped the the coffee filter from the top of the balcony so that we could gather the data in order to find the terminal velocity. This was done by capturing videos of the filters falling. Then we found the final velocity by graphing the Position vs. Time for each of the coffee filters. Then we used the terminal velocities to graph the Speed vs. Force which gave us the k and n for the equation Air Resistance Force = kv^n. We then used Excel in order to verify our k and n values which proved to be correct. Overall, this lab proved to be successful since we came to the same results and there could be some uncertainty because interpreting the data can not be 100% accurate. All in all, the air resistance force is directly related to the speed as proved by the lab results.