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Terminal Velocity vs Area of a Falling Object

Submitted by Rich on Tue, 12/04/2018 - 00:22

Terminal Velocity Introduction

The effect of mass on the terminal velocity of an object falling in air is commonly done using basket coffee filters.  But how could we study the effect of area on the terminal velocity of a falling object?  One way to do this is to use PocketLab Voyager and its range finder along with a single piece of cardstock as the object to be dropped.  In this lesson, students discover a relationship between area and terminal velocity using CloudLab.  Results are  compared to a common model for air resistance, also known as drag.

One way to do this is to use a single sheet of 8½ x 11 inch cardstock. Cardstock was found better than thinner paper since, being stiffer, it kept its shape better while falling. While holding the cardstock with its surface horizontal, it is released about one meter above a PocketLab Voyager that is lying on the floor.  The card stock is then folded in half and released a second time. It is again folded in half and released a third time.   It is folded in half a final time and then released. Data from the PocketLab app allows determining the terminal velocity in each of these four cases. We have controlled mass since the entire mass of a single sheet is used for each case, and we have controlled the shape, at least to the extent that we have a flat surface in each case. If we take the area after the three folds to be 1 arbitrary unit of area (AUA), then we have cardstock areas of 1, 2, and 4 and 8 AUA, as shown in Figure 1. 

Cardstock sizes for the terminal velocity experiment
Figure 1 - cardstock folds with areas 1, 2, 4,  and 8 AUA (Arbitray Units of Area)

For each of the four areas, it is a good idea to drop the cardstock several times and use the average terminal velocity from the drops.  Figure 2 shows a graph of height (above the floor) vs. time for the case of dropping the cardstock before any folds have been done, i.e., for 8 AUA.  The data rate was set to 25 points/s.  The average slope of the region from 1.8 s to 2.4 s represents the desired terminal velocity.

Typical graph of position vs time
Figure 2

Studying Terminal Velocity with CloudLab

If you are not familiar with CloudLab, it is strongly suggested that you read the short article "The CloudLab Model" before proceeding.  It will give you an overview of the structure of CloudLab.

We now take you on a tour of CloudLab via a series of screenshots taken after our experiment was completed.  Each screen shot is accompanied by a short discussion explaining the screenshot.

The top of Figure 3 shows the title of our Lab Report.  Our experiment consisted of 4 runs.  Each of the runs was given a name representing the area of the card stock piece being dropped. Each of the runs consisted of three trials.  You can see that it is quite easy to add additional runs by clicking the green "Add Another Run" button.  If we click on the down-arrow to the far right of the "Area = 8 AUA" run, we can then see the details of that run.

CloudLab Lab Report information
Figure 3

Figure 4 shows a summary of the three trials for the "Area = 8 AUA" run.  The "Trial Results" are the terminal velocities of the pieces of card stock for which the area is 8 AUA.    At the far right of each trial are three icons.  The pencil icon allows the student to enter the trial result.  The middle icon opens up the graph of the data recorded by Voyager for that run.  The trash can icon lets you discard a faulty trial.

The Run "Area = 8 AUA" details
Figure 4

Let's now look at the graph of the data recorded in Trial 1 of the "Area = 8 AUA" run, as shown in Figure 5.  The shaded region represents the time during which the piece of card stock was falling toward Voyager.

Highlighted graph
Figure 5

After dragging the highlighted region, we obtain a zoomed in graph of just that highlighted region, as shown in Figure 6.  We click on the "Data Analysis" button and select the linear option for curve fitting.  We see that the slope of the best fit equation is -1.16 m/s.   Since we are interested only in the magnitude, we key in 1.16 as the "Trial Result".

This process is repeated for each of the three trials in each of the four runs.  After this we are ready to set up the Results Table.

Data analysis of Trial 1 of "Area = 8 AUA"
Figure 6

Figure 7 shows the results table in the Data Analysis Toolkit.  The independent variable is keyed in as Area (AUA).  The values 8, 4, 2, and 1 are then keyed in for the card stock areas for each of the runs.  The dependent variable is keyed in as Terminal Velocity (m/s).  In this experiment, rather than using Manual Entry for the dependent variable, we click the radio button for "Trial Mean".  For each of the four runs, CloudLab then calculates the mean value for the three trial results in each run and fills in those values automatically for us.

Results table for our experiment
Figure 7

We can then work with the Results Graph, shown in Figure 8.  A Bar graph will display by default, but we click on "Line" since we want a line graph.  It is seen that the axes are automatically numbered and labeled in accordance with the Results Table of Figure 7.  It appears that there is some kind of an inverse power relationship terminal velocity and area.

The Results graph of our experiment
Figure 8

We are now ready for the data analysis of the results graph of Figure 8, so we click on "Data Analysis" in the lower left corner of the figure.  Upon clicking the "Data Analysis" button, we select power curve fitting  and obtain what is shown in Figure 9.  We see that the power -0.32 provides a fit with an r-squared value very close to 1.  This power is very close to an inverse cube root relationship, with terminal velocity inversely proportional to the cube root of the area.  Different student groups are likely to get somewhat different powers but all should be inverse power relationships.  

Power relationship graph between terminal velocity and area
Figure 9

Finally, we add a "Section" to our Lab Report  for which we give the title "Conclusion", as shown in Figure 10.  We have included a photo of the 8 AUA card stock being held above Voyager just before dropping the card.  Although we haven't done so, we could also include a You Tube video.  The ability to add hyperlinks is also planned.  Additional "Sections" can be added by clicking on the "Add Another Section" button.

Conclusion section of our experiment
Figure 10

 

Relationship between terminal velocity and area
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