Not So Free Fall Measuring the Terminal Velocity of Coffee Filters

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Constance Elaine Tate
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1 Not So Free Fall Measuring the Terminal Velocity of Coffee Filters When solving physics problems or performing physics experiments, you are often told to ignore air resistance or assume the acceleration is constant or that energy is conserved. Air resistance does affect the motion of all objects, sometimes in a negligible fashion but at other times in a profound manner. In this activity, the effects of air resistance cannot be ignored. Hopefully, you will not meet too much resistance while doing this activity. When an object falls in air, the air exerts an upward force called the drag force. As the speed of the falling object increases, so does the drag force. When the drag force equals the weight of the object, the object stops accelerating and falls with a constant terminal velocity. Objects such as feathers, pieces of paper, and coffee filters exhibit this behavior and reach their terminal velocity rapidly because they have a large surface area and a small mass. Repeated experiments with objects falling in air show that sometimes the drag force is proportional to the velocity, and sometimes it is proportional to the square of the velocity. The direction of the drag force in either case is opposite the direction of the velocity. The drag force can be expressed mathematically as F drag bv or 2 F drag cv where the constants b and c are the drag coefficients, and depend on the size and shape of the object as well as the density of the air through which it moves. As an object falls in air, there are two forces acting on it: the weight due to gravity (mg) and the air resistance or drag force (F drag ). The speed of the falling object increases until the drag force is equal and opposite the weight. At this point, the velocity is no longer changing but remains constant. This velocity is called the terminal velocity (v 1 ). Depending on whether the drag force is proportional to the terminal velocity or the square of the terminal velocity, the terminal velocity relates to the mass as mg bv t or 2 mg cv t Therefore, if we can measure the terminal velocity of an object whose shape stays the same while its mass can be increased, plotting mass versus the terminal velocity and plotting mass versus the square of the terminal velocity will allow us to determine the appropriate relationship. Notice that according to both models a relative velocity of 0 m/s implies that the object feels no drag force due to the air. Therefore, (0, 0) is a theoretical point for both models. 73

2 Not So Free Fall Physics You will use paper coffee filters to answer the key question for this activity: How does air resistance or the drag force change with terminal velocity? These basket-shaped filters are close to ideal because they combine very low mass with a relatively large cross-sectional area, and they can be easily nested inside one another to provide a wide variety of masses with identical shapes. By dropping one or more coffee filters toward an ultrasonic motion detector and studying the resulting motion graphs (position vs. time and velocity vs. time), you will be able to determine the terminal velocity. Using an analytic balance, the mass (as well as the weight) of the filters can also be determined. PURPOSE You will observe the effects of air resistance (the drag force) on falling coffee filters, and determine how the terminal velocity of a falling coffee filter is affected by mass. MATERIALS Each lab group will need the following: calculator, TI graphing 5 coffee filters computer LabQuest meter stick ruler, clear metric sensor, motion detector Safety Alert! Use caution when standing on a step ladder or lab table. 74

3 PROCEDURE 1. Determine the mass of the six coffee filters using an electronic balance. (You may find the total mass of all six filters together and then divide by six to find the average mass of one filter.) Record these measurements in the data table on your student answer page. Coffee Filter(s) Motion Detector or Figure 1 Figure 2 2. Support the motion detector about 2 m above the floor, pointing down, as shown in Figure 1, or hold the motion detector directly above the coffee filter, as shown in Figure If your motion detector has a switch, set it to Normal. 4. Connect the motion detector to DIG 1 of LabQuest and choose New from the File menu. If you have an older sensor that does not auto-id, manually set up the sensor. 5. Place a coffee filter in the palm of your hand and hold it about 0.5 m under the motion detector if using the setup of Figure 2, or hold the filter as high as possible directly over the motion detector if using the setup of Figure 1. Do not hold the filter closer than 0.15 m. 6. Start data collection. After a moment, release the coffee filter directly below (or over) 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. 75

4 Not So Free Fall Physics 7. Examine your position graph. 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. 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. 8. 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. Because 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. Tap and drag your stylus across the linear region to select it. Choose Curve Fit from the Analyze menu. Select Linear as the Fit Equation. Record the slope in the data table (velocity in m/s). Select OK. 9. Repeat steps 3 8 for two, three, four, five, and six coffee filters. (Optionally, extend to seven and eight filters but be sure to use a sufficient fall distance so that a clear velocity can be measured.) 76

5 DATA AND OBSERVATIONS Determining the filter s mass and weight. Total Mass of 6 Filters (kg) Data Table 1 Average Mass of 1 Filter (kg) Average Weight of 1 Filter (N) Determining the mass, weight, and terminal velocity of each filter combination. Data Table 2 Number of Coffee Filters Mass (kg) Weight (N) Terminal Velocity (m/s)

6 Not So Free Fall Physics ANALYSIS 1. How does the drag force or air resistance change with the terminal velocity? Plot a graph of drag force (weight) vs. terminal velocity. Depending on whether the drag force is proportional to the terminal velocity or the square of the terminal velocity, the terminal velocity relates to the mass as mg bv t or 2 mg cv t 78

7 The shape of the graph will give the value of the exponent. For example, if the graph is a straight line, the drag force is directly proportional to the velocity and so the exponent is 1. If the graph is a parabola, then the relationship is a quadratic in which case the exponent is 2. If the graph is a cubic function, then the exponent is What does your graph indicate the exponent to be? 79

8 Not So Free Fall Physics 3. Plot a graph of drag force (weight) versus the square of terminal velocity. 4. From the analysis of your data, what is the equation that fits your six data points? 80

9 CONCLUSION QUESTIONS 1. A small object of mass m is located near Earth s surface and falls from rest in Earth s gravitational field. Acting on the object is a resistive force of magnitude kmv 2, where k is a constant and v is the speed of the body. a. On the figure shown, draw and identify all of the forces acting on the body as it falls. b. Using Newton s second law ( F net ma), determine the terminal speed v t of the object. c. On the axes provided, draw a graph of the speed v as a function of time t. 81

10 Not So Free Fall Physics 2. On the figures provided, draw and identify all the forces acting on the coffee filter as it falls from rest in Earth s gravitational field. Describe how the magnitude and direction of the net force on the coffee filter is changing over this time period (from the moment of release until the terminal velocity has been reached). Moment of release Before terminal After terminal 3. On the axes provided, draw a graph of the speed v of the coffee filter as a function of time t. 82

11 Even objects that fall freely for only a few meters reach terminal velocity. If this were not true, then falling objects such as rain drops could have severe consequences, and hail stones would be even more destructive. How fast it falls depends on the size and weight of the raindrop: the heavier, the faster. At sea level, a large raindrop about 5 mm across (housefly size) falls at the rate of 9 m/s (20 mph). Drizzle drops less than 0.5 mm across (i.e., salt-grain size) fall at 2 m/s (4.5 mph). A raindrop starts falling and then picks up speed because of gravity. Simultaneously, the drag of the surrounding air slows the drop s fall. The two forces balance when the air resistance just equals the weight of the raindrop. Then the drop reaches its terminal velocity and falls at that speed until it hits the ground. This simple view neglects updrafts, downdrafts, evaporation, air density, and other complications. The air resistance depends on the shape of the raindrop, the cross-sectional area presented to the airflow, and the raindrop s speed. Most drops are fairly round the small ones spherical, larger ones flattened on the bottom by the airflow. At high speeds, the air resistance increases with the square of the velocity. 4. If we assume that objects falling in air continued to accelerate without reaching terminal velocity, calculate the following: a. Calculate the speed of a raindrop if the raindrop were in freefall. Assume the rain falls from 1000 m above the surface of Earth, and that air friction has no effect. b. If 26.7 m/s is approximately equal to 60 mph, what is the speed of the raindrop in miles per hour? 83

Not So Free Fall Not So Free Fall

Physics Not So Free Fall Not So Free Fall Measuring the Terminal Velocity of Coffee Filters About this Lesson In this activity students will observe the effects of air resistance on falling objects. In