Purpose: Determining the relationship between air resistance force and speed, and seeing if we can model the tall of an object including air resistance.
Theory: Once an object reaches terminal velocity while falling, the force of friction from the air and the weight of the object are the net force, which is equal to the mass and its acceleration. We can use this relation to create an expression for air resistance with velocity as a term, and can test how accurate our model is by comparing the predicted velocity vs. actual measured velocity. Hopefully with only 10% of error.
Procedure: Take five coffee filters, weigh them to determine mass, drop one from a certain height, record the fall, place another coffee filter atop the previous one, and repeat dropping and recording until you have used all five filters. Load videos into logger pro, in order to determine velocity of each trial run, record velocity, create a velocity vs mass times gravity, use it to create an equation/model for air resistance, test model on excel and compare it to the recorded velocity to determine how accurate it was.
*First we dropped a coffee filter from a balcony.
*Loading the video into logger pro, we were able to determine the velocity by plotting points at each frame of where in relation to the balcony (roughly 1 meter) the filter was while falling.
*We repeated the process, always adding another coffee filter to increase the mass, five times.
*We determined the terminal velocity of each run, recorded the mass of each run, and calculated the force of mass times acceleration due to gravity of each run.
Recordings
|
V (m/s)
|
M (kg)
|
F (N)
|
|
0.9578
|
0.0008947
|
0.008768
|
|
1.323
|
0.001758
|
0.01723
|
|
1.605
|
0.002684
|
0.0263
|
|
1.922
|
0.003579
|
0.03507
|
|
2.253
|
0.004473
|
0.04384
|
*We created a velocity vs force of mass*gravity graph, since the only two forces acting on the falling filter are air resistance (with velocity as a factor in the equation) and the filter mass times acceleration due to gravity, we hope to see some type of relation between the two on a graph.
Graph
*The graph is exponential, and when fitted to give an equation it gives us the force of the mass times gravity is equal to the velocity times a constant taken to a power (mg=Av^B).
*A and B are constants, and represent factors that would effect air resistance, like surface area and shape and friction of the air.
*The value for A given from logger pro is 0.01105 and B is 1.725, and v is the terminal velocity of a falling coffee filter
*We now test our model using excel.
*Our model is the net force(the mass as it accelerates downwards) will equal the force of weight minus the force of air resistance
(ma=mg-Av^B)
*In time increments of only .001 seconds, the change in velocity will be the acceleration determined from our model multiplied by the time increment.
*Eventually the changes in velocity will become so minute, the velocity will be virtually unchanged by them, this is our terminal velocity.
*For a single filter of mass 0.0008947kg
Results
|
t=
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0.001
|
||||
|
A=
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0.01105
|
||||
|
B=
|
1.725
|
||||
|
m=
|
0.0008947
|
||||
|
t (s)
|
a (m/s^2)
|
∆v (m/s)
|
v (m/s)
|
∆x (m)
|
x (m)
|
|
0
|
9.8
|
0
|
0
|
0
|
0
|
|
0.001
|
9.79576796
|
0.0098
|
0.0098
|
0.0000098
|
0.0000098
|
*Scrolling down
|
t
(s)
|
a
(m/s^2)
|
∆v
(m/s)
|
v
(m/s)
|
∆x
(m)
|
x
(m)
|
|
0.485
|
0.00216469
|
2.2074E-06
|
0.8743945
|
0.00087439
|
0.36720737
|
|
0.486
|
0.00212285
|
2.1647E-06
|
0.87439666
|
0.0008744
|
0.36808177
|
|
0.487
|
0.00208182
|
2.1229E-06
|
0.87439879
|
0.0008744
|
0.36895617
|
|
0.488
|
0.00204158
|
2.0818E-06
|
0.87440087
|
0.0008744
|
0.36983057
|
|
0.489
|
0.00200212
|
2.0416E-06
|
0.87440291
|
0.0008744
|
0.37070497
|
|
0.49
|
0.00196342
|
2.0021E-06
|
0.87440491
|
0.0008744
|
0.37157938
|
*At close to 0.486 of second the velocity to four significant figures "settles" at 0.8744 m/s, all further changes to velocity will be so miniscule, it can be argued that the filter is now at terminal velocity.
*For the first run the "predicted" velocity was 0.8744 m/s, actual 0.9578, error 8.7%
*For the first run the "predicted" velocity was 1.304 m/s, actual 1.323, error 1.4%
*For the first run the "predicted" velocity was 1.653 m/s, actual 1.605, error 2.9%
*For the first run the "predicted" velocity was 1.953 m/s, actual 1.922, error 1.6%
*For the first run the "predicted" velocity was 2.222 m/s, actual 2.253, error 1.4%
Conclusion
*My model worked very well, all "predicted" terminal velocities were only off by less than 10%, so I am satisfied with the values of A and B, and the formula Av^B as a model for the force of air resistance.
*Perhaps the greater error for the first run was because of how more "malleable" a single coffee filter is, its surface area and shape could change easier while it fell vs. multiple coffee filters holding each others shape in place.
*The whole process might be time consuming, as the constants A and B change depending on the shape of the object, its surface area, and air temperature effecting air density. All contribute to air resistance, so the entire experiment would have to be done again for another object of differing shape and multiple outside temperature.
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