13 Mar 2017: Lab 4: Modeling the Fall of an Object with Air Resistance

Lab 4: Modeling the Fall of an Object with Air Resistance

Joel Cook

Nina Song

Lab performed March 13, 2017

In this experiment, we are attempting to develop a model for calculating the air resistance and terminal speed of an object.

Introduction: The air resistance an object experiences depends on the speed, shape, and mass of an object. By using video capture, we can analyze the speed, acceleration, distance, and time an object travels. By analyzing coffee filters being dropped, we are able to calculate the terminal velocity of the coffee filter. In the second part of this experiment, we use Microsoft Excel to numerically determine the terminal velocity of coffee filters and compare the results against the video capture to determine if the model created is acceptable. We expect that the air resistance can be calculated using the equation
F=kv^n, where F is air resistance force, k is a constant for shape and area of the object, and v is speed. 

Apparatus/Procedure: In this lab, we used a laptop computer and logger pro to capture and analyze the video of coffee filters being dropped. We captured video of one filter being dropped, two filters together being dropped, and repeated for a total of five filters dropped together. The filters were stacked together to keep the shape of the object the same but change the mass for each trial. The videos were analyzed to plot the position of the filter versus time. A linear fit of the last portion of the data shows the terminal velocity of the coffee filter, which is the slope of the line through the points. A graph of Air Resistance Force versus speed was created showing one through five filters. A power fit of this data generates a model for calculating the Net Force on an object. 

For the second part of the experiment, we used Microsoft Excel to numerically analyze the data using the model created in the first part of the lab. The spreadsheet uses time, speed, net force, acceleration, mass of the object, and the equation determined part one, to calculate the terminal velocity. We are able to compare the values observed in experiment to the values determined from the model to determine how good our model is. In the spreadsheet, time used was increments of 0.05 seconds, Net Force is mass times acceleration minus Air Resistance Force (k*v^n), acceleration starts at g and is Net Force divided by mass, change in speed is acceleration calculated times time increment, and speed is previous speed plus the calculated change in speed. By adding rows to the spreadsheet, we can see the point when velocity stops increasing and acceleration is zero. This velocity is the terminal velocity, or the highest velocity the object can reach in free fall.

Data: 

The photo below shows the data generated from the video capture for one coffee filter dropped. The terminal velocity shown by the slope of the linear fit of the data is 1.263 m/s.

The photo below shows the data generated from the video capture for two coffee filters dropped. The terminal velocity shown by the slope of the linear fit of the data is 1.679 m/s.

The photo below shows the data generated from the video capture for three coffee filters dropped. The terminal velocity shown by the slope of the linear fit of the data is 2.106 m/s.

The photo below shows the data generated from the video capture for four coffee filters dropped. The terminal velocity shown by the slope of the linear fit of the data is 2.508 m/s.

The photo below shows the data generated from the video capture for five coffee filters dropped. The terminal velocity shown by the slope of the linear fit of the data is 2.728 m/s.

The photo below shows the data generated by plotting the terminal velocities versus mass. A power fit of the curve is shown.

The equation for air resistance is now F=kv^n is now F = 0.0005989 * v ^ 1.958
The uncertainty in k is +/- 0.00005879 (N*s/m) and the uncertainty in n is +/- 0.1086.

For the second portion of the lab, we determined the terminal velocity based on numerical analysis using Microsoft Excel. The photo below shows the result for terminal velocity of one coffee filter being dropped. The terminal velocity was calculated as 1.2115 m/s.

The photo below shows the result for terminal velocity of two coffee filters being dropped. The terminal velocity was calculated as 1.7261 m/s.

The photo below shows the result for terminal velocity of three coffee filters being dropped. The terminal velocity was calculated as 2.1213 m/s.

The photo below shows the result for terminal velocity of four coffee filters being dropped. The terminal velocity was calculated as 2.4593 m/s.

The photo below shows the result for terminal velocity of five coffee filters being dropped. The terminal velocity was calculated as 2.7562 m/s.

The table below summarizes the results for the video analysis versus the numerical analysis:

# of Filters	Mass of Filters (kg)	Terminal Velocity from Video Analysis (m/s)	Terminal Velocity from Numerical Analysis (m/s)	Difference
1	8.72E-04	1.263	1.2115	0.0515
2	1.74E-03	1.679	1.7261	-0.0471
3	2.62E-03	2.106	2.1213	-0.0153
4	3.49E-03	2.508	2.4593	0.0487
5	4.36E-03	2.728	2.7562	-0.0282

For all cases, the value for terminal velocity calculated in Microsoft excel was within 0.1 m/s of the value calculated from the video analysis method.

Conclusion: Our model we created for determining air resistance and modeling terminal velocity end up being very very accurate but there are variables in the lab that are difficult, or impossible, to control. When filming the coffee filters, the lens of the camera is turning a cone shaped image into a flat image which will shorten the distances at the edges of the image; therefore the most accurate velocities are when the coffee filter was in the center of the image. It is not possible to account for this in the data. In addition, the video analysis requires clicking on a slightly blurry coffee filter that was difficult to see and impossible to click identically for each frame of each video. Patience and persistence helped to make the video analysis as accurate as possible but uncertainty in the data exists for this portion. Finally, when scaling the video during analysis, there is uncertainty in the accuracy of properly selecting the right distance for the scale. These possible uncertainties were considered and mitigated as much as possible by careful analysis of the videos.