Lab 4: Capturing Air Resistance on Tape!

Today, we set up sneaky webcams to try to catch naked air work its magic against coffee filters. We hope to see air push up against them. So we all march to the Design Technology building like a tabloid firm and hang around the staircase. But what we're doing is not so insidious, so we say. All we're doing is finding the hidden relationship between air and speed...

The process is that by dropping 1-5 coffee filters off that balcony (look up!), and capturing it with a camera, we could use video analysis software to discover its terminal velocity--the velocity of an object in free fall caused by the air resistance working against gravitational force. We also measure some reference point in the same depth so we know how many pixels the camera records per some unit of measurement. And we could do this while annoying a few bystanders, shooting two birds with one stone.

 

Because air resistance counters gravity, we know that it is a force, such that:

F_resistance = kvⁿ

And therefore, we find should find the terminal velocity and the force. Let's get the velocity out of the way first. The velocity is found by going through the video frame by frame and noting the change in position by the coffee filters. LoggerPro has a function to do just that; it also plots a position graph with that data, and we use a linear best-fit line to get an equation. We expect that the filters will fall some distance before it hits terminal velocity by air resistance, but we're not interested in the acceleration under normal gravity in the beginning. We want to find the slope of the linear displacement, which occurs after terminal velocity. This took some manipulation in LoggerPro.

As they all look exceedingly similar with slightly different slopes, here is just one example of a graph with the best fit line. We have 5: one for each number of coffee filters.

Here's what a data table looks like. The graph shown above is for 2 filters. Here's the data table for 3 filters:

Now we take all the tables generated by LoggerPro, and export it into a standardized CSV (comma separated values) file, which will be easier (in my opinion) to look at, and possibly work with in the future. Although we could import CSV directly into a spreadsheet program, we did not do this for this lab. Here's a photo showing what the CSV looks like, although as you can probably surmise, the data table for higher number of filters take too much space.

Anyways, from the data table, and consequently the graph, we get the terminal velocities for each number of filters:

• 1 Coffee Filter: 0.9222 m/s
• 2 Coffee Filters: 1.272 m/s
• 3 Coffee Filters: 1.641 m/s
• 4 Coffee Filters: 2.129 m/s
• 5 Coffee Filters: 2.525 m/s

Now that we got our independent variable, v on the formula above, we need to find F_resistance. Since it's a force, which is mass * gravity, we need to find the weight of the coffee filters in kilograms, and then multiply them by the gravitational constant to get force in Newtons. With a small digital scale, we find the weight of a single coffee filter is 1.035 g, which is 0.010143 N.

Using a curved fit line, we find k and n based on experimental data.

k = 0.01420 ± 0.00185

n = 1.390 ± 0.1071

Since we have working numbers now, we are ready to model the fall of the filters in Excel. This time, we look for the terminal velocity by looking at the velocity when acceleration goes to 0 m/s². We begin with a time interval of 30 Hz, because that is the refresh rate of the camera.

We can see from the spreadsheet that acceleration seems to be moving too quickly, and doesn't approach 0 m/s² close enough, and as such won't provide us with very accurate data. Therefore, we change the time interval to 200 Hz.

There, better. Pull the data down and find the terminal velocity.

As acceleration approaches 0 m/s², the velocity seems to settle around 0.785 m/s. When we look at our experimental data, we see that the terminal velocity we recorded is 0.9222 m/s. Not very close. However, we also see that our first sample point didn't really fall on the fit-line. If we choose a point that is closer to the line, then our modeled data should be closer to the experimental data (since the model is based on the fit-line, its data points should conform to it).

We choose the point for 2 filters, which sits just above the line, which means we expect that our experimental data will be just under the model. So we change the mass in the acceleration equation to reflect the mass of 2 filters, from 0.00135 kg to 0.0027 kg.

And, pull the data down, once again...

The terminal velocity with 2 filters seems to be approaching 1.292 m/s. While with 1 filter, the difference between our experimental and modeled data is approximately 0.14 m/s, with 2 filters, the difference is only about 0.02 m/s. And it is a little larger than our experimental data as predicted!

So in conclusion, we've proved that by looking at the model, we can predict things about what happens in experiments. The model allows us to interpolate to any mass and find its terminal velocity.