Lab 4: Modeling the Fall of an Object Falling with Air Resistance
Amy, Chris, and John
March 13, 2017
Today's lab was essentially determining the relationship between air resistance force and speed.
Theory: The expectation that air resistance force on a particular object depends on the object's speed, its shape, and the material it is moving through. By calculating the object's terminal velocity we can find what the air resistance was. Through finding an object's terminal velocity the (mg of the object equals air resistance) because there is no acceleration and the net force on the falling object is zero.
Through the power law equation given below, our group modeled the air resistance of the coffee filters.
In this power law formula we determined
k takes into account the shape and area of the object, and
v is the velocity. We do not know what
k or
v are, so we enacted scenarios where we knew what the force of air resistance was and the velocity the coffee filter reached. By recording all these data we were able to graph all these scenarios and find the values of the unknown variables of the power law.
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| Dropping coffee filters in front of a black fabric |
Apparatus and Procedure: For this specific lab we used about 6 coffee filters, laptop, a long sheet of black fabric. We placed our laptop by the stairs facing the drape where our professor set up the drape above the floor hanging all the way to the ground. We set up our Logger Pro video capture in the Design Technology building. As the professor dropped the coffee filters we recorded the falls. After all the coffee filters were dropped from the balcony, we went back to the classroom and began plotting points in our videos. On the blue drape there were two pieces of tape giving us the ratio of one meter we could adjust in our video capture to better match the position of each coffee filter, we used this to measure the distance our coffee filter covered every 3rd frame, this was to reduce the amount of points we had to plot in our video.
When the coffee filter is first dropped it accelerates downward with gravitational force (mass*gravity) and as it speeds down the air resistance force is introduced, such that when the force of gravitational matches the air resistance force, the coffee filter reaches terminal velocity. This is because the net force acting upon the coffee filters is zero, so it is no longer accelerating downward.
On LoggerPro, we plotted the position of the filter which gave us the Position vs. Time graph where we could find terminal velocity by doing a linear fit towards the end of the graph in order to find where the air resistance force equaled gravitational force, because that would mean the velocity of the coffee filter is constant.
By doing this multiple times (1-6 filters) we were able to record the terminal velocities as well as the air resistance force. We graphed this data and on a graph of Air Resistance Force vs. Terminal Velocity and found the linear fit for our data and this gave us the values for
k and
n. Later we compared our values with the values we gathered from doing the mathematical procedure.
Measured Data and Graphs: To find the mass of each coffee filter Professor Wolf measured the mass of 50 coffee filters to rule out the uncertainty of one filter. 50 coffee filters weighed 43.9 g +/- 0.01 g. We divided this number by 50 in order to find the mass of a single filter. After conversions the mass of a coffee filter came out to be 0.000878 +/- 0.0002 kg.
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| 1 coffee filter dropped, The slope or terminal velocity: -1.136 m/s |
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2 coffee filters dropped, slope or velocity: -1.715 m/s
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| 3 coffee filters, slope or velocity: -2.035 m/s |
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| 4 coffee filters dropped, slope or velocity: -2.439 m/s |
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| 5 coffee filters dropped, slope or velocity: -2.578 m/s |
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| 6 coffee filters dropped, terminal velocity or slope: -2.786 m/s |
The next graph is the power law fit for six of our coffee filters being dropped. That gave us an equation to figure out the values of
k and
n. The values we got for k is 0.005303 with uncertainty of +/- 0.0008248 and the value for n is 2.195 with an uncertainty of +/- 0.1675.
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| Power Fit for our Air Resistance Force and Velocity graph. |
Values we obtained for for terminal velocity:
| Number
of filters |
Terminal
Velocity (m/s) |
| 1 |
1.24670191 |
| 2 |
1.70964575 |
| 3 |
2.05650554 |
| 4 |
2.34449423 |
| 5 |
2.59537045 |
| 6 |
2.8201448 |
| Number
of filters |
Terminal
Velocity (m/s) |
Terminal
Velocity from Graph (m/s) |
|
| 1 |
1.24670191 |
1.366 |
|
| 2 |
1.70964575 |
1.715 |
|
| 3 |
2.05650554 |
2.035 |
|
| 4 |
2.34449423 |
2.439 |
|
| 5 |
2.59537045 |
2.578 |
|
| 6 |
2.8201448 |
2.786 |
|
|
|
|
|
Conclusion: By comparing the values calculated vs. the values from our video plotted graph we notice that the values are fairly close. This is probably due to miscalculations from our graphs and the plotted points from our videos causing propagated uncertainty. While we were plotting the points in our video it was a bit difficult to find where the coffee filter ended up. Because the coffee filter was moving faster than our camera could record, and it sometimes fell towards the brighter side of the screen, then we were not able to plot it in our graph with much accuracy. Our points are the human error causes of uncertainty, but we also have the calculated values for k and n may also have uncertainty even though our correlation of 0.9942 was high.
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