9-Mar-2015/11-Mar-2015: Measuring air resistance

PURPOSE

The purpose of this experiment was to find the relationship between air resistance and terminal velocity.

PROCEDURES

Part 1

Figure 1: Some of us dropped the filters from the balcony

Figure 2: While others tracked the filters motion from downstairs

We began this experiment by gathering five coffee filters and heading over to the Design Technology building. Then, we proceeded to drop them from the balcony as shown in Figure 1 and captured the motion (Figure 2) with our laptops. We started with one filter and added another after each drop. We maintained the first filter as the bottom filter to keep a constant surface going against the air resistance throughout the entirety of the experiment. After we finished dropping all five filters, we returned to the classroom and used video motion analysis in Logger Pro to record their displacements during each time interval. We graphed these displacements against time and implemented a Linear Fit onto the linear portion of the graph. The reason for this was that within the curved portion of the graph, the velocity was increasing. Therefore, we concluded that when the graph became straight, the filter stopped accelerating and had reached its terminal velocity. Consequently, we used the slope expressed in the linear fit as the terminal velocity. The graph for five filters is illustrated in Figure 3 below. 

Figure 3: Displacement vs time graph/terminal velocity

From these terminal velocities, we were able to come up with a model for the the air resistance equation F = k * v^n. We did this by graphing the air resistance force against the terminal velocities and applying an Auto Fit to the resulting graph. We found the air resistance force by applying Newton's second law of motion in the y-direction. Since the filters were at their terminal velocities, their accelerations were zero. Therefore, we came to the conclusion that the air resistance force must equal the filters' weights. In order to find the weight of one filter, we first had to measure its mass. We scaled its mass by finding the mass of fifty filters and dividing the value by fifty. Then, we multiplied the mass of the filter by the acceleration due to gravity to find its weight. We assumed that each filter had the same mass. so we just multiplied the weight of one filter by the number of coffee filters to find the weights of each number of coffee filters. The resulting graph of plotting the air resistance forces against the terminal velocities is displayed below (Figure 4).

Figure 4: Air resistance force vs terminal velocity graph

From this graph, we were able to find the k and n values of the air resistance equation from the A and B values illustrated above. believed A and B were equal to k and n, respectively, because the equation of the graph above (Figure 4) had the same format as the air resistance equation, F = k * v^n

Part 2

In the second part of the lab, we wanted to accomplish the exact opposite of what we did in part 1. Our goal of this portion of the lab was to use the k and n values found from part 1 and find the terminal velocities of each mass. We accomplished this task by setting up an Excel file with these constants as part of the constraints. In addition, we constructed six columns. The first column was time (t). The second column was average acceleration (a). The third and fourth column were the change in velocity (Δv) and the average velocity (v), respectively. The fifth was displacement (Δx) and the sixth was position (x). We found the Δv values by multiplying the a values by Δt, which was set at 0.002. We then found v by adding Δv to each subsequent v. Next, we calculated Δx by multiplying v with Δt. Finally, we found x by adding Δx to each subsequent x. The spreadsheet is shown in Figure 5 below.

Figure 5: Mathematical model used for finding terminal velocities

From Figure 5, it can be seen that the filter's acceleration is approaching zero at 0.548 seconds. Therefore, we concluded that the velocity corresponding to this time should be the terminal velocity. We found the terminal velocity to be 2.007 m/s, which was close to the experimental value of 2.019 m/s. The comparisons of the experimental terminal velocities and theoretical terminal velocities can be seen below (Figure 6).

Figure 6: Data set comparing experimental and theoretical values

CONCLUSION

This lab was useful in helping us further understand air resistance and how it affects the motion of objects. The lab was even more effective in doing so because of the fact that we solved for terminal velocities in two different ways: experimentally and by using mathematical models. 

The reason why the terminal velocities measured in the experiment were different from the ones found using mathematical models is due to the various sources of error. One of these sources of error is the way the coffee filters fell down. We noticed that sometimes the filters were swaying back and forth as they were falling down. We can see that this had a bigger affect on the lower number of filters because their masses were smaller and less stable. Another source of error could have been the video analysis part of the lab. This step was very problematic because it was difficult to track the motion of the filter accurately using video analysis. First of all, it was virtually impossible to click the same spot on the coffee filter when we were trying to mark its location every frame.

Regardless of these sources of error, this lab was beneficial because it allowed to develop a deeper understanding of what is going on in the physical world.