Find out the relationship between air resistance and speed of a falling object

Find out the relationship between air resistance and speed of a falling object.

Introduction: We see certain relationship between air resistance of a falling object and its speed, and in this lab, we will do an experiment to try to define this relationship.

Apparatus:

 We use this camera that can connect to a computer to film the falling motion of coffee filters, and use logger pro to create v vs. t graphs of each trial.

In this lab, we use coffee filters as the falling objects because it has a small weight and large surface, which can help us see the change air resistance make to the falling motion. and we can easily change the weight of the falling object by stacking more  than one coffee filters together.




Theosis:
We expect the magnitude of the air resistance force on a particular object depends on the object's speed. We expect the equation to look like: F = kv^n
Procedure:
1. We first filmed the free fall motion of different numbers of coffee filters, and mark the position of coffee filters in each split second. Logger pro then produced a position vs. time graph with the points we marked.
2. We will get total of 5 graphs since we did 5 trial with different weight, and we will find the slope of the part of the graph that is straight because the straight part of the graph indicate the coffee filter has reached its terminal speed. The slope will indicate the terminal speed of coffee filters.
3. We will then create a resistance force vs. terminal speed graph, where the resistance force will be the same as the weight of the coffee filters.
4. We will then use the graph to find n and k in the equation: F=kv^n.

Data and Analysis:
Following is the v vs. t graph of our 5 trials

First trial with one coffee filter, which has a total weight 0.010143N, the terminal velocity where resistance force =total weight is 1.489 m/s

Second trial with two coffee filters, which has a total weight 0.020286N, the terminal velocity where                             resistance force =total weight is 1.651 m/s 

Third trial with three coffee filters, which has a total weight 0.030429N, the terminal velocity where                                     resistance force =total weight is  1.844 m/s

Fourth trial with four coffee filters, which has a total weight 0.040572N, the terminal velocity where                         resistance force =total weight is 2.146 m/s

Fifth trial with five coffee filters, which has a total weight 0.05071N, the terminal velocity where                               resistance force =total weight is 2.236 m/s

We can get this following chart from the above graphs:

Coffee Filters	Terminal Velocity(m/s)	Force(N)
1	-1.489	0.01014
2	-1.651	0.02028
3	-1.844	0.03042
4	-2.146	0.04057
5	-2.236	0.05071

The example of calculate:(20 coffee filters=20.7 grams)
1 coffee filters: F=mg=(20.7/20)/1000 *9.8=0.01014

Then we make Resistance Force vs. Velocity graph:

Then we use the fit feature in logger pro and find out that A = k = 0.003913 and B= n =3.154.

Thus, our equation for relationship between air resistance force and traveling velocity is:

F = 0.003913v^3.154

In the Resistance Force vs. Velocity graph above, we found that our second trial and fifth trial is closer to the fit curve than other points, so we decided to use Excel to model our 2nd and 5th trial, and find out how much is the % error.

To model falling motion of each trials, we make the time interval to be 1/30 s. ΔV=a*Δt, V=V previous+ΔV,a=g-(kv^n)/m => a=9.8-0.003913*v^3.154/0.02028, Δx=(V previous+V)/2 *Δt, X=X previous+Δx.

2nd trial

The chart we modeled for the falling motion of our second trial indicate that the terminal speed of the falling object should be 1.685m/s, and our experimental value is 1.651 m/s^2.

% error = (1.651-1.685)/1.685 *100%= 2.0%

The chart we modeled for the falling motion of our fifth trial indicate that the terminal speed of the falling object should be 2.253m/s, and our experimental value is 2.236 m/s^2.

% error = (2.236-2.253)/2.253 *100%=0.75%

Conclusion:

The relationship between air resistance and speed of a falling object:              Fresistance= 0.003913v^3.154

The relationship equation we found for the air resistance depend on velocity ends up to work very well. The percent error is very low when plug in theoretical number compare to the experimental value. The possible reason for the 2% and 0.75% error is that every single coffee filter has a slightly different weight, but we count every coffee filter's weight with the average of 20 of them; the way we marked the position of coffee filter in each split second of the falling motion in the video is not very precise because of the low resolution of the video and our eye can not sharply determine the real position of the coffee filter in each split second.