Procedure Air Resistance

Air Resistance

Air Resistance

  Graphical Analysis 13

Air Resistance

Related Video: https://youtu.be/FYgez2qcmgs

Introduction

When you solve physics problems involving free fall, often you are told to ignore air resistance and to assume the acceleration is constant. 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 a meter 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 describe it with the term terminal velocity, or vT. The paper reaches terminal velocity very quickly, but on a short drop to the floor, the baseball does not.

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 Fdrag = –bv or Fdrag = –cv2. 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 –cv2. At terminal velocity, the downward force is equal to the upward force, so mg = –bv or mg = –cv2 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

vT  m      or       vT2  m

roportional

roportional

or

If we plot mass versus vT or vT2, 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.

objectives

· Observe the effect of air resistance on falling coffee filters.

· Determine how air resistance and mass affect the terminal velocity of a falling object.

· Choose between two competing force models for the air resistance on falling coffee filters.

Materials

Chromebook, computer, or mobile device

Graphical Analysis 4 app

Go Direct Motion

5 basket-style coffee filters

Preliminary questions

Warning: Read the Introduction section above before answering the preliminary questions

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?

2. Sketch your prediction of a graph of the velocity vs. time for one falling coffee filter.

3. If a filter is moving at a constant velocity, what do you know about the forces on the filter?

Procedure

1. Support the motion detector about 2 m above the floor, pointing down, as shown in Figure 1.

Figure 1   

2. Set up the motion detector and Graphical Analysis.

a. Launch Graphical Analysis.

b. Connect the motion detector to your Chromebook, computer, or mobile device.

c. If two graphs are displayed, click or tap View, , and choose 1 Graph.

3. 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 0.25 m.

4. Click or tap Collect to start data collection. After a moment, 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.

5. Examine your position vs. time graph.

a. If the motion of the filter was too erratic to get a smooth graph, you will need to repeat the measurement. With practice, the filter will fall almost straight down with little sideways motion.

b. To collect data again, simply start data collection when you are ready to release the filter. Continue to repeat this process until you get a smooth graph. Note: The previous data set is automatically saved when you click or tap Collect.

6. The velocity of the coffee filter can be determined from the slope of the position vs. time graph. At the start of the graph, there should be a region of increasing slope (increasing velocity), and then the plot 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 (vT) during that time.

a. Select the data in the linear region.

b. Click or tap Graph Tools, , for the position vs. time graph.

c. Select Linear as the curve fit, and click or tap Apply.

d. Record the slope in the data table (a velocity in m/s).

7. Repeat Steps 3–6 for two, three, four, and five coffee filters. (Optionally extend to six, seven, and eight filters, using a sufficient fall distance so that a clear velocity can be measured.)

Data Table

Number of filters	Terminal velocity vT (m/s)	(Terminal velocity)2 vT2 (m2/s2)
1
2
3
4
5

Analysis

1. To help choose between the two models for the drag force, plot terminal velocity vT vs. number of filters (which represents mass). On a separate graph, plot vT2 vs. number of filters. Use Graphical Analysis or graph paper. Scale each axis from the origin (0,0).

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 vT  m. Or, if the drag force is proportional to the square of velocity, then vT2  m. 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 –cv2) 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. 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?

Extensions (VOID)

1. Make a small parachute and use the motion detector to analyze the air resistance and terminal velocity as the weight suspended from the parachute increases.

2. Draw a free body diagram of a falling coffee filter. There are only two forces acting on the filter. Once the terminal velocity vT has been reached, the acceleration is zero, so the net force, ΣF = ma = 0, must also be zero

ΣF = –mg +bvT = 0       or        ΣF = –mg +cvT2 = 0

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 vT first.)

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