Air Resistance
When you solve physics problems involving free fall,
often you are told to ignore air resistance and to assume the acceleration is
constant and unending. In the real world, because of air resistance, objects do
not fall indefinitely with constant acceleration. One way to see this is by
comparing the fall of a baseball and a sheet of paper when dropped from the
same height. The baseball is still accelerating when it hits the floor. Air has
a much greater effect on the motion of the paper than it does on the motion of
the baseball. The paper does not accelerate very long before air resistance
reduces the acceleration so that it moves at an almost constant velocity. When
an object is falling with a constant velocity, we prefer to use the term terminal velocity, or
Air resistance is sometimes referred to as a drag force. Experiments have been done with a variety of objects falling in air. These sometimes show that the drag force is proportional to the velocity and sometimes that the drag force is proportional to the square of the velocity. In either case, the direction of the drag force is opposite to the direction of motion. Mathematically, the drag force can be described using F_{drag} = –bv or F_{drag} = –cv^{2}. The constants b and c are called the drag coefficients that depend on the size and shape of the object.
When falling, there are two forces acting on an object:
the weight, mg, and air resistance, –bv or
–cv^{2}. At terminal velocity, the downward force is
equal to the upward
force, so mg = –bv or
mg = –cv^{2}, depending on whether the drag force follows the first or second
relationship. In either case, since g
and b or c are constants, the terminal velocity is affected by the mass of
the object. Taking out the constants, this yields either
v_{T} _{} or v_{T}^{2} _{}
If we plot mass versus v_{T} or v_{T}^{2}, we can determine which relationship is more appropriate.
In this experiment, you will measure terminal velocity as a function of mass for falling coffee filters and use the data to choose between the two models for the drag force. Coffee filters were chosen because they are light enough to reach terminal velocity in a short distance.
1. Hold a single coffee filter in your hand. Release it and watch it fall to the ground. Next, nest two filters and release them. Did two filters fall faster, slower, or at the same rate as one filter? What kind of mathematical relationship do you predict will exist between the velocity of fall and the number of filters?
2. If there was no air resistance, how would the rate of fall of a coffee filter compare to the rate of fall of a baseball?
3. Sketch a graph of the velocity vs. time for one falling coffee filter.
4. When the filter reaches terminal velocity, what is the net force acting upon it?
Procedure
1. Connect the Motion Detector to DIG/SONIC 2 of the LabPro. 2. Support the Motion Detector about 2 m above the floor, pointing down, as shown in Figure 1. 3. Start LoggerPro on a laptop, and Open the Physics with Computers folder, and the Air Resistance (Exp 13) file. A graph will appear, with the vertical axis (distance) scaled from 0 to 3m. The horizontal axis (time) is scaled from 0 to 4 s. 4. Place a coffee filter in the palm of your hand and hold it about 0.5 m under the Motion Detector. Do not hold the filter closer than 0.4 m. 5. Click _{} to begin data collection. When the Motion Detector begins to click, release the coffee filter directly below the Motion Detector so that it falls toward the floor. Move your hand out of the beam of the Motion Detector as quickly as possible so that only the motion of the filter is recorded on the graph. 
Figure 1 
6. If the motion of the filter was too erratic to get a smooth graph, repeat the measurement. With practice, the filter will fall almost straight down with little sideways motion.
7. The velocity of the coffee filter can be determined from the slope of the distance vs. time graph. At the start of the graph, there should be a region of increasing slope (increasing velocity), and then it should become linear. Since the slope of this line is velocity, the linear portion indicates that the filter was falling with a constant or terminal velocity (v_{T}) during that time. Drag your mouse pointer to select the portion of the graph that appears the most linear. Determine the slope by clicking the Linear Regression button, _{}.
8. Record the slope in the data table below (velocity in m/s).
9. Repeat Steps 4 – 8 for two, three, four, and five coffee filters.
Number of filters 
Terminal Velocity v_{T} (m/s) 
(Terminal Velocity)^{2} v_{T}^{2} (m^{2}/s^{2}) 
1 


2 


3 


4 


5 


Analysis
1. To help choose between the two models for the drag force, plot terminal velocity v_{T} vs. number of filters (mass). On a separate graph, plot v_{T}^{2} vs. number of filters.
2. During terminal velocity the drag force is equal to the weight (mg) of the filter. If the drag force is proportional to velocity, then v_{T} _{}. Or, if the drag force is proportional to the square of velocity, then v_{T}^{2} _{}. From your graphs, which proportionality is consistent with your data; that is, which graph is closer to a straight line that goes through the origin?
3. From the choice of proportionalities in the previous step, which of the drag force relationships (– bv or – cv^{2}) appears to model the real data better? Notice that you are choosing between two different descriptions of air resistance—one or both may not correspond to what you observed.
4. Draw a free body diagram of a falling coffee filter. There are only two forces acting on the filter. Once the terminal velocity v_{T} has been reached, the acceleration is zero, so the net force, ĺ F = ma = 0, must also be zero
_{} or _{}
depending on which drag force model you use. Given this,
sketch plots for the terminal velocity (y
axis) as a function of filter weight for each model (x axis). (Hint: Solve for v_{T} first.)
5. How does the time of fall
relate to the weight (mg) of the
coffee filters (drag force)? If one filter falls in time, t, how long would it take four filters to fall, assuming the
filters are always moving at terminal velocity?
For your class presentation:
 Aim for 8  10 minutes for the presentation. Everyone on the team should have a chance to talk in front of the class for at least 2 minutes. Be sure to introduce the team members in your presentation.
 Remember some teams will not have done this lab, so you’ll need to provide a bit of background.
 Be able to show your graphs visually, as either a handout or a computer file that can be projected to the class, and explain the results.
 Be able to discuss uncertainties in your measurements and how they affect your results.
 Please turn in one copy of your data, graphs, and answers to the questions above along with any handout you produce for your short presentation. This is NOT intended to follow the formal lab report format.