Lab 4: Modeling the Fall of an Object Falling with Air Resistance
Chris Ceron, Amy Chung, John Choi
Purpose
To determine the relationship between air resistance force and speed.
Theory/Introduction
We expect that the air resistance force on a particular object is dependent on the object's speed, its shape, and the material it is moving through. This can be modeled through the following power law:
| k and n are unknown variables |
The k term takes the shape and area of the object into account.
To find the unknown variables, we had to create several scenarios where we calculate the terminal velocities within each scenario. The objects used in the scenario had to have the same size and shape, but also needed different masses. Once these objects reached their terminal velocity, the downward pull of gravity exactly balanced the upward force of air resistance.
In our lab, we used coffee filters:
To collect our data, we used the video capture feature within LoggerPro, and created six different cases by dropping multiple coffee filters stacked together in the Design Technology Building. Our cases went from dropping 1 coffee filter to 6 coffee filters.
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| Black drape was used to clearly see the falling coffee filter |
Through the videos we captured, we were able to create graphs for each scenario that allowed us to plot position vs. time graph that would allow for us to calculate the terminal velocity.
Procedure
The professor was in charge of dropping the filters and adjusting each case by adding a coffee filter. The black drape had two pieces of blue tape that were each one meter apart from each other. This served as a scale for our position vs. time graph. In order to calculate the gravitational force, we needed to know the mass for each coffee filter. If we were to take the mass of an individual coffee filter, our uncertainty would be large. To reduce the uncertainty, the professor took the mass of 50 coffee filters, which allows for us to divide both the mass and the uncertainty of an individual coffee filter.
Once we recorded each case, we used LoggerPro to plot the position of the coffee filter after every three frames. This created a position vs. time graph. In each graph, we could calculate the terminal velocity by taking a linear fit in the more linear part of the graph. Each graph would become more linear within the end of the graph because the coffee filters would reach terminal velocity and move at a constant distance per time as a result of air resistance force balancing the gravitational force.
Position vs. Time Graphs for Each Scenario:
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| Graph for one coffee filter: Terminal velocity is -1.136 m/s |
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| Graph for two coffee filters. Terminal velocity is -1.715 m/s |
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| Graph for three coffee filters. Terminal velocity is -2.035 m/s |
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| Graph for four coffee filters. Terminal velocity is -2.439 m/s |
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| Graph for five coffee filters. Terminal velocity is -2.578 m/s |
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| Graph for six coffee filters. Terminal velocity is -2.786 m/s |
Once we calculated the terminal velocity for each case, we created a Force of Air Resistance vs. Terminal velocity graph that allowed for us to find our unknown variables k and n.
Results:
k = A = 0.005303 ± 0.0008248
n = B = 2.195 ± 0.1675
Numerical Calculations
Our next approach was to model the fall of the object with air resistance and predict the terminal velocity in each case through Microsoft Excel. To set this up, we used the numerical approach given in our lab handout
Once we set up our excel sheet, we used a time interval of 1/125th of a second and found where our velocity remaining constant.
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| Terminal velocity for one coffee filter |
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| Terminal velocity for two coffee filters |
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| Terminal velocity for three coffee filters |
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| Terminal velocity for four coffee filters |
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| Terminal Velocity for five coffee filters |
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| Terminal velocity for six coffee filters |
Conclusion
Comparison of the two terminal velocities we calculated through LoggerPro and Excel are relatively close, but are different due to some uncertainties in our measurements and calculations. When plotting the position of the coffee filter as it was falling, we are subject to human error as our plots may not be as accurate as we would like. In the video recording, the coffee filters would occasionally fall outside of the black drape, making it harder to accurately plot a position within that frame. When plotting our terminal velocities to calculate for k and n, our correlation was .9942. Although this is a good correlation, it is still not the highest and most accurate.
Our ratio of uncertainty in our n value also caused a difference between the two terminal velocities, as the ratio of dn/n was approximately 7.6%. In order to calculate a more accurate terminal velocity through LoggerPro, we would need to plot more accurate points in order to find a more accurate k and n value.
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