16-March-2015: Coffee Filter Lab

The purpose of this lab was to find a relationship between the air resistance force and terminal speed of falling coffee filters. Along with this, we were to input this relationship into Excel to try to model the fall of our coffee filters with air resistance.

Part 1: Gathering Data
The goal of the first part of this lab was to find the terminal velocity and air resistance force for some bound coffee filters. We will be basing our data off the assumption that F(resistance)=kv^n, where F(resistance) is the air resistance force, k is a term that accounts for the shape and area of an object, v is the terminal velocity of the object, and n is some unknown numerical value. F(resistance) and v are the more practical values we can find.

Procedure:
The way we conducted this lab was with 5 coffee filters. We went to the Design Technology building at Mt. SAC to conduct the experiment. There, we would drop the desired amount of coffee filters, record the falling with video capture, and find the terminal velocity of the filters using LoggerPro. There were several steps to accomplishing this:

Fig. 1
This was the point of view of the webcam
with which we were going to perform
our videos capture.

1. We brought a laptop with us to the Design Technology building. We set it up on the stairs in such a way that its webcam would be facing the balcony from which we would be dropping the coffee filters. We opened up a LoggerPro video capture window. One person would be controlling the laptop.
2. We had a second person go to the balcony, where they would first drop one coffee filter alone, the two coffee filters stuck together, then three coffee filters stuck together, and so forth all the way to five coffee filters. (Idea: adding more filters increase both the mass, air resistance force, and terminal velocity). This person would also be holding a meter stick visible to the camera. (see Fig. 1)
3. Once we were all set up, we dropped the coffee filters and recorded the fall for all five runs. Once completed, we returned back to class to analyze the data. 

Obtaining Data:

On the laptop, we had 5 different videos for five different drops. In order to find the terminal velocity of the coffee filters, we would need to create a position vs time graph for the fall. To do this, we scaled the video and inputted a position axis into it. The meter stick that was held during the drop was used to scale the video. (Note: we flipped the position axis 180 degrees. This way, the graph we would be obtaining would be positive and rising as the object physically fell.) No that the video was prepared, we began to plot points onto the video of where the coffee filter was every 1/30th seconds. We did this all the way down as it fell, which resulted in our position graph for that specific filter drop (see Fig. 2). We repeated this for all he coffee filters we dropped.

Fig. 2
There was special preparation for the video we captured to ensure that we got fairly accurate results.

Once we had our position vs time graph for each of the filters, we gave a linear fit for the portion of the graph that seemed to have the most consistent slope. This consistent slope would indicate that the coffee filter(s) had reached their terminal velocity. The linear fit would provide us with the slope of that line, which in turn was the terminal velocity (see Fig. 3).

Fig. 3
The only important part of this position graph is where the slope starts to even out. This is where we will find our
terminal velocity.

We now had to find the air resistance force for each of the coffee filters. To do this, we simply used Newton's 2nd Law. When the filters reached their terminal velocity, the net force acting on the system was now zero. With only air resistance force and weight acting on the system, we could then find that F(resistance)=mg. Thus, we needed to find the mass of a single coffee filter. Since each filter is so light, we measured the mass of 50 coffee filters, and then divided by 50 to find the mass of 1 filter.One coffee filter had an approximate mass of 0.9 g. We now knew the air resistance force and terminal velocity for each of the filter drops. With this data, we created a table of air resistance force vs terminal velocity (see Fig. 4) We used this data table to create a graph of air resistance force vs terminal (see Fig. 5) We input a power fit through the points which yielded us the result of F=0.01348v^1.886.

Fig. 4
As we added more and more filters,
the mass of the system rose, as did the
air resistance force and terminal
velocity.

Fig. 5
This linear fit gave us our unknowns of
k and n.

Part 2:

Now that we had our mathematical model, we wanted to try to predict the terminal velocity of the coffee filters by creating a model in Excel. To do this we opened up a new spread sheet and inputted our known values of k, n, m(mass of 1 coffee filter), ∆t, and g(9.8m/s^2). Beneath this, we created 6 columns: t(time), ∆v(change in velocity), v(velocity), a(acceleration), ∆x(change in position), and x(position). The initial value of each of these variables was zero, except for acceleration, which started at 9.8 m/s^2. For our time column, we inputted our equation for time, which was previous t + ∆t. For our ∆v, our equation was the product of the previous acceleration and ∆t. For our velocity column, our equation simply added the previous velocity with ∆v. For our acceleration, we derived our expression to be a=g-((kv^n)/m) (see Fig. 6). Our equation for ∆x was simply average velocity multiplied with ∆t. Finally, for our position column, we simply added ∆x to the previous position.

Fig. 6
We had to derive an expression for acceleration that we could input into Excel.

Our goal was now to fill down until acceleration was relatively zero and the velocity was consistent. This constant velocity would be the terminal velocity for that specific coffee filter. To test the other coffee filters, we would simply adjust the mass. (see Fig. 7-1 and Fig. 7-2)

Fig. 7-1
This was the general look for our spread sheet modeling the terminal velocity of our coffee filters.

Fig. 7-2
Once the acceleration and ∆v became zero, we knew we had reached our terminal velocity, which
for one coffee filter was 0.81 m/s.

We adjusted the spread sheet five times, each time changing the mass to the new amount of coffee filters we wanted to model. Overall, Excel was able to model us fairly good results. Our percent error from our obtained value for terminal velocity and Excel's value is seen in Fig. 8.

Fig. 8