11-March-2015: Air Resistance Lab

Lab 4: Modeling the fall of an object falling with air resistance

Part 1: Determining the relationship between air resistance force and speed

Purpose: To determine the relationship between air resistance force and speed

Assumption: Air resistance force on a particular object depends on the object's speed, its shape, and the material it is moving through: F= k*v^n

Data Collection: For this experiment, we used the video capture feature on LoggerPro.

Procedure:
The procedure for this lab was fairly simple. We had to capture video for 1,2,3,4, and 5 coffee filters falling from the balcony of building 13. First, we did a test run in the classroom to familiarize ourselves with capturing video and inputting the data into a position vs. time graph.  Then we headed over to the Design Technology building to complete the experiment. The indoor balcony gave us the height necessary so that the falling coffee filters would reach terminal velocity before landing on the ground. After taking video for the five trials, we headed back to the classroom to analyze the results.

Data:
Using LoggerPro, we tracked the coffee filter's position over time. The data was then inputted into a P v T graph. We then took a linear fit of the plot graph and calculated the terminal velocity. In this case the slope of the graph equals the coffee filter's terminal velocity. This step was done for 1,2,3,4, and 5 coffee filters.

 

 

P v T graph: 1 coffee filter



P v T graph: 1 coffee filter

Terminal Velocity:

• 1 filter: Slope= 0.816 m/s
• 2 filters: Slope= 1.216 m/s
• 3 filters: N/A
• 4 filters: Slope= 1.650 m/s
• 5 filters: Slope= 1.865 m/s

There was an error in the video capture of three coffee filters falling. So the data for three coffee filters will be forfeited.

Only applies at terminal velocity

Finding values for k and n:

As stated above F= k*v^n. In order to determine our missing values, we must plot F-air vs. V (terminal).

From the free-body diagram of a coffee filter, we see that F-air = mg, where g= 9.8 m/s and m= mass of coffee filter.
In this case 1 coffee filter= .000926 +/- .002 kg.

• We weighed 50 coffee filters.                                 For a total of  46.3 +/- .1 g
• Mass of 1coffee filter = 46.3g /50

Now, we plotted this data; where
x(column):Terminal velocity of coffee filters
y(column):Calculated weight of each filter= F-air

 As you can see in the graph above, we only plotted four points because the data for three coffee filters was not available.

Finally, we solved for k and n by taking a power fit of the graph.
From the graph: k= 0.01279 +/-0.0006846 , n= 2.042+/-0.09734

Part 2: Modeling the fall of an object including air resistance

Purpose: To apply the mathematical model developed in part 1 to predict the terminal velocity of the various coffee filters

From part 1, we found that F-air resistance= k*v^n, where F-air= mg.
Therefore Fnet=ma=mg-k*v^n
We then solved for acceleration, as seen below. In theory when a=0, the falling object is at terminal velocity.

From here, we used to Excel to model the fall of an object with air resistance.
First, we established six constants so that we could easily change the mass of the filters and not have to set up the columns again on a new spreadsheet.
The constants are:

• g= 9.81 m/s
• n= 2.042
• m= Mass of filter(subject to change)
• k=0.01279
• Delta t= 0.0001

Excel Setup:

• First Column (Set to 0): Time= Change in time (.0001)
• Second Column (Set to 0): Change in velocity= a1*change in time
• Third Column (Set to 0): Velocity= v1+ change in velocity
• Fourth Column (Set to 9.8): a= 9.8- (k/m)*v^n
• Fifth Column (Set to 0):  Change in x= (v2+v1)/2*Change in time
• Sixth Column (Set to 0): x= x1+change in x

If setup properly, the spreadsheet should look like this:

Numerical Data for 1 coffee filter



•  Notice how velocity is increasing and acceleration is decreasing.

Eventually, the acceleration reaches zero. At this point the filter has reached terminal velocity.

Numerical Data for 1 coffee filter

 As you can see from the spreadsheet, when a=0, the terminal velocity= 0.846 m/s. So the terminal velocity for 1 coffee filter is 0.846 m/s
This process was repeated for the various coffee filters.

Numerical Terminal Velocity:

• 1 filter: v= 0.816 m/s
• 2 filters: v= 1.19 m/s
• 3 filters: N/A
• 4 filters: v= 1.67 m/s
• 5 filters: v= 1.860 m/s

 Terminal Velocity from Part 1:

• 1 filter: v= 0.816 m/s
• 2 filters: v= 1.21 m/s
• 3 filters: N/A
• 4 filters: v= 1.650 m/s
• 5 filters: v= 1.865 m/s

 As you can see the terminal velocities from Part 1 and Part 2 are fairly similar. This proves that our model works.  The velocities are not exactly the same due to the uncertainty in the k and n values. Nonetheless, if we tinkered with the numbers long enough; the velocities would match. However, there may be some limitations to our model because our data sample is simply not big enough. There can also be a few sources of error. For example, the scaling in the videos for the falling coffee filters could be off. The video capture could also show error, if the timing was off between the person capturing video and the person dropping the coffee filter. In all there will always be room for error. For the time being, we proved that our model works and accurately predicts the terminal velocity of various coffee filters.