The purpose of this experiment was to find an equation that provides the relationship between the speed of a falling object and the air resistance force on the object.
Apparatus:
The apparatus of this experiment consisted of a computer that contains the LoggerPro software. LoggerPro was used for video capture and the creation of graphs and data tables. Five brown coffee filters and a meter stick were also used. The video capture program of the LoggerPro software was used to record the coffee filters in free fall, and then to track their position relative to time. The position and time data were then used to create a graph to find the velocity of the freely falling coffee filter(s). The meter stick was used to give the LoggerPro software a reference for the distance that the coffee filter(s) fall in an interval of time.
Abstract (Part 1):
In order to create a model for the force of air resistance on the object, it was first guessed that the force depends on velocity, since the air particles under the freely falling object are accelerated by the object from "rest" in the downwards vertical direction to the speed of the falling object. Therefore, it was first guessed that the air resistance force is equal to to the mass of the air particles in the way multiplied by the acceleration, or change in velocity over change in time.
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However, it was then deduced that if, for example, the speed of a falling object is doubled, then twice as many air particles are being accelerated to the speed of the object. Therefore, the guess was modified such that the air resistance force is proportional to the square of velocity, or
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But, even though more particles are accelerated if the velocity is increased, the mass of the air is not proportional to speed in that many air particles just move out of the path of the falling object. Therefore, it was finally decided that the air resistance force depends not only on the falling object's speed, bu also the shape of the object and the material it is moving through. In addition, the force is not directly proportional to the velocity, yet there is an unknown relationship between them. With these principles taken into account, the following equation was derived:
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where A is a constant that takes into account the size and shape of an object, V is the velocity of the object, and n is a constant that provides the "proportionality" of the relationship between velocity and air resistance force.
Procedure:
Once all of the materials for this experiment were obtained, a test for the video capture program on LoggerPro software was performed by dropping the coffee filters in class. Once the test was done and the video capture confirmed to be correct, the experiment was moved to the Design Technology Building, or building 13. This experiment was done indoors so as to minimize any other forces acting on the freely falling coffee filters (windage, for example). Then, using the meter stick as a reference for distance, video capture was used to record the fall of the coffee filters. First, one coffee filter was dropped and its fall recorded. Then, the fall of two coffee filters put together was recorded. Then the falls of three, then four, and then finally five coffee filters were recorded. Once the falls were recorded, a position versus time graph was formed for each of the five falls by using LoggerPro. An origin was selected and the position of the coffee filters was noted every few frames. The origin was selected as the coffee filters' start of fall at the top of the building 13 balcony. Using the position and time data, the graphs were formed. Using those graphs, the terminal velocities of each of the five trials were found by forming a linear fit of the data points at the near end of the graph, since the filters reach terminal velocity a bit before the end of their fall. Taking the slope of those points, the terminal velocity can be found. These graphs are shown below:
Figure 3|: Position vs time graph of the fall of one coffee filter
Using the linear fit at the near end of the points, the slope of the linear fit provides the expected terminal velocity. The terminal velocity of the fall of 1 coffee filter is around 0.83 m/s.
Figure 4: Position vs time graph of the fall of two coffee filters
The fall of two coffee filters has a terminal velocity of 1.22 m/s.
Figure 5: Position vs time graph of the fall of three coffee filters
Three coffee filters, when dropped together, had a terminal speed of 1.44 m/s.
Figure 6: The position vs time graph of four coffee filters falling together
The fall of four coffee filters had a terminal velocity of 1.75 m/s
Figure 7: Position vs time graph of the fall of five coffee filters
The fall of five coffee filters had a terminal speed of 1.85 m/s.
With the terminal velocities of all five trials, a graph was constructed that displayed the force of air resistance vs velocity. The force of air resistance can be found easily since at terminal speed the falling object is not accelerating; its force of air resistance is equal to its weight. To determine the force of air resistance, the mass of 50 brown coffee filters were measured, and then that mass was divided by 50 to get the mass of each filter. Then, that mass was multiplied by the value of gravity to obtain the force of air resistance. The mass of 50 brown coffee filters was measured to be 46.3 g ± 0.1 g. Dividing by 50, the mass of one filter was found to be 0.926 g ± 0.002 g. With this value, the following formula was derived to find the air resistance force:
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where n is the number of coffee filters. Using this formula, the air resistance force for all five trials were found and the data table shown below was created.
Figure 8: Data table of terminal velocity versus air resistance force
The last data point had to be crossed out since the change in velocity from four to five coffee filters was not significant, skewing the graph. With the above data table, the following graph was made:
Figure 9: Air resistance force vs velocity graph
As can be seen, the equation of the above graph fits the model equation derived that relates force of air resistance and velocity (see formula 4). From the graph, the constant A was found to be 0.009587 ± 0.001727. The value of n was found to be 3.568 ± 0.3429.
Abstract/Procedure (Part 2):
Figure 10: Solving for acceleration (the change in velocity with respect to time)
In order to test the model created in Part 1 of the experiment, Excel was used to model a falling object with air resistance containing the same mass, A, and n values, and then the terminal velocities found in Excel were compared to the ones found in part 1. In the above figure, the acceleration of a freely falling object at terminal speed was calculated using Newton's Second Law (F = ma), where k is equal to A.
Figure 11: Predicted layout for the model test on Excel
As seen in the figure above, six columns were made in Excel. The initial values for time, change in velocity, velocity, change in position, and position were all zero. Acceleration, however, began ta the value of gravity. The time interval between each data row (Δt) was 0.01 sec. Δv for the new row was then found by multiplying the acceleration of the previous row by Δt = 0.01 sec. "v" was then found by adding the change in velocity of that row and the velocity of the previous row. Acceleration was the only value based on the derived model. Using the equation for acceleration derived in figure 10, "a" was found by subtracting the value of gravity by (k/m)vⁿ, using the velocity of that row. Δx was then found by taking the average of the sum of the velocity of that row and the previous row, and then multiplying that value by Δt = 0.01 s. Lastly, "x" was found by adding the value of x from the previous row and Δx from that row. Using Excel, this process was repeated for all five coffee filters until v began to not change by much. This indicates that terminal velocity of the falling object has been reached. The Excel data tables for each of the five trials are found below:
From Excel, the terminal velocity of 1 coffee filter was 0.977 m/s.
The terminal velocity of two coffee filters was 1.187 m/s.
The terminal velocity of three coffee filters was 1.329 m/s.
The terminal velocity of four coffee filters was 1.441 m/s.
Figure 16: Excel data table for 5 coffee filters
The percentage difference of the terminal velocities for one coffee filter is 16.65%. The percentage difference of the terminal velocities for two coffee filters was 3.04%. The percentage difference of the terminal velocities for three coffee filters was 8.09%. The percentage difference for four coffee filters was 19.25%. Lastly, the percentage difference for five coffee filters was 18.84%.
Conclusion:
All of the percentage differences are in the range of the uncertainty displayed in the A and n values, roughly 10 - 20%. The percentage differences for two and three filters were even lower than that; however, there is large deviations between the percentage differences, and the uncertainty is very large. This shows that the model created to relate force of air resistance and velocity was not accurate enough to explain the real effects of the world on a falling object.
Most of the uncertainty in this experiment was caused by the poor terminal velocity values measured, which all traces back to the equipment used in the experiment. The equipment used was not good enough for an experiment such as this one that needs high accuracy. To lower the uncertainty in this experiment, a laptop with better processing power, a newer version of LoggerPro software, a camera that records faster (at around 60 frames per second) and has sharper recording quality (higher megapixels), and darker colored coffee filters (so that it is easier to see in the video capture, resulting in more accurate noting of position) of higher quality (so that they fall more efficiently i.e. less change in horizontal position, more accurate representation of air resistance on a falling object, etc.) would be needed.
In addition, in order to lower any human error (e.g. noting wrong positions on LoggerPro, dropping the coffee filters incorrectly so as to cause them to sway, fall faster that actual, etc), more time should have been given to video capture the falls of the coffee filters. More time would have allowed for a greater number of trials performed, testing the accuracy of the video capture software after each trial recorded, and allowing for more detailed video captures that could have resulted in more accurate values. Lastly, a better reference should have been used for the distance the coffee filters had moved. For example, a meter stick could have been taped onto the side of the balcony to allow for better recording of the reference. Also, the height of someone or something could have been used as well since it is easier to note something larger than a meter stick from a farther view.
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