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 most introductory physics labs friction of all sorts is ignored or only mentioned as a possible source of error. However, air friction plays a significant role in free-fall and the concept of terminal velocity is an important concept which should be explored and understood by students. Objective The students will: observe the effect of air resistance on falling coffee filters determine how the terminal velocity of falling filters is affected by surface area and mass choose between two competing force models for the air resistance on falling coffee filters Level Physics Common Core State Standards for Science Content LTF Science lessons will be aligned with the next generation of multi-state science standards that are currently in development. These standards are said to be developed around the anchor document, A Framework for K 12 Science Education, which was produced by the National Research Council. Where applicable, the LTF Science lessons are also aligned to the Common Core Standards for Mathematical Content as well as the Common Core Literacy Standards for Science and Technical Subjects. Code Standard Level of Thinking (LITERACY) RST.9-10.3 (MATH) A-CED.2 (LITERACY) RST.9-10.7 (MATH) F-IF.6 Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Depth of Knowledge Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 67
Not So Free Fall Physics (MATH) S-ID.6a (MATH) S-ID.6c (MATH) F-LE.2 (MATH) S-ID.8 (MATH) S-ID.7 (MATH) F-LE.5 Code Standard Level of Thinking (LITERACY) W.4 (MATH) A-CED.4 (MATH) N-Q.1 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a linear function for a scatter plot that suggests a linear association. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Compute (using technology) and interpret the correlation coefficient of a linear fit. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Interpret expressions for functions in terms of the situation they model. Interpret the parameters in a linear or exponential function in terms of a context. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm s law V = IR to highlight resistance R. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Depth of Knowledge 68 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall Connections to AP 1. Newtonian mechanics, B. Newton s laws of motion, 2. Dynamics of a single particle (second law), e. The effect of fluid friction on the motion of a body, (1) Find the terminal velocity of a body moving vertically through a fluid, (2) Describe qualitatively, with the aid of graphs, the acceleration, velocity of displacement of such a particle. Materials Each lab group will need the following: calculator, TI graphing 5 coffee filters computer LabQuest meter stick ruler, clear metric sensor, motion detector Additional teacher materials: balance Assessments The following types of formative assessments are embedded in this lesson: Assessment of prior knowledge Guided questions Assessment of Process Skills Visual assessment of procedures and analysis of data The following additional assessments are located on the LTF website: Activity Quiz: Newton s Laws Physics Assessment: Kinematics 1 and 2 D Physics Assessment: Newton s Laws of Motion 2011 Physics Posttest, Free Response Question 1 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 69
Not So Free Fall Physics Teaching Suggestions As an introductory demonstration for students, you may wish to take a single basket-type coffee filter and drop it from a height of about 2 m (standing on a step-ladder is the best way to do this). Pick up the filter and crumple it into a compact ball, then drop it again and note how much faster it falls. Drop it simultaneously with a small wooden or steel ball. This shows the importance of the surface area. Air resistance is the result of collisions of the filter s leading surface with air molecules. Now take a stack of coffee filters and a single coffee filter and drop them simultaneously. This illustrates the importance of the weight. A heavier coffee filter with same area as a lighter one must fall faster to reach terminal velocity. Hence, more massive filter stacks have a greater terminal velocity and fall further in the same time. At terminal velocity, the weight of an object equals its air resistance. It is important to keep your dropping pattern constant throughout the experiment. The students may have difficulties getting coffee filters to fall straight down, but after a few tries the experiment should work. To collect enough data points, the motion detector needs to be at least 1.5 m from the release point of the coffee filters. By placing the motion detector on the floor and dropping the filters directly over the detector, a slightly smoother graph can be obtained than by mounting the motion detector above the floor and releasing the filters from below the detector. However, the graph displayed will be the height of the filters versus time, and some students may find the resulting negative and decreasing velocity graph difficult to comprehend. Therefore, it may be advisable to drop the filters away from the detector so that the velocity increases and reaches a maximum (terminal) velocity. 70 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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 the plot should become linear. Because the slope of this line represents velocity, the linear portion indicates that the filter was falling with a constant or terminal velocity. The graph shown displays the filter moving away from the detector, as would be the result in the Figure 2 setup in the procedure. This experiment works equally well if you videotape the falling filters against a measuring stick and play the video back frame by frame. This allows you to measure the distance the coffee filters fall each 30 1 of a second. Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 71
Not-So Free Fall Measuring the Terminal Velocity of Coffee Filters DATA AND OBSERVATIONS Determining the filter s mass and weight Data Table 1 Total Mass of 6 Filters (kg) Average Mass of 1 Filter (kg) Average Weight of 1 Filter (N) 0.004868 kg 0.0008110 kg 0.007952 N Determining the mass, weight, and terminal velocity of each filter combination Number of Coffee Filters Mass (kg) Data Table 2 Weight (N) Terminal Velocity (m/s) 1 0.000811 0.007952 0.786 2 0.001622 0.015896 1.087 3 0.002433 0.023843 1.466 4 0.003244 0.031791 1.723 5 0.004055 0.039739 1.930 6 0.004866 0.047687 2.244 T E A C H E R P A G E S
ANALYSIS 1. How does the drag force or air resistancee change withh 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 terminall velocity, the terminal velocity relates to the mass as mg bv t or mg 2 cv t Drag Force vs. Terminall Velocity 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 velocityy 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 3. T E A C H E R P A G E S 2. What does your graph indicate the exponent to be? The relationship between the drag force and the terminal velocity is definitely not linear. The graph is a parabola, indicating a quadratic relationship between the drag force (weight of the coffee filters) and the terminal velocity. Hence, the exponent is 2.
3. Plot a graph of drag force (weight) versus the square of terminal velocity. Drag Force vs. Terminal Velocity Squaring the terminal velocities yields a straight line. The slope is the constant, N b = 0.0095 (m 2 m/s) T E A C H E R P A G E S 4. From the analysis of your data, what is the equation that fits your six data points? F drag = 0.0095(v t ) 2, where the constant depends on the cross-sectional area of the filters, the smoothness of the filter, and the density of thee air.
CONCLUSION QUESTIONS 1. A small object of mass m is located near Earth s surface and fallss 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. Drag Force Weight Force b. Using Newton s second law ( F net ma ), determine the terminal speed v t of the object. At terminal velocity, the net force is 0 and thee acceleration is 0. The net force is the differencee between the weight force and the drag force. 2 mg kmv ma 0 Fnet ma g kv v t 2 g k c. On the axes provided, draw a graph of the speed v as a function of time t.. Speed vs. Time T E A C H E R P A G E S
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). Initially, the drag force is negligible and the net force is directed downward. As the speed of the filter increases, the drag force gets larger but the net force is still directed downward. Once the filter has reached terminal velocity, the acceleration is 0 and the drag force equals the weight force, so there is no net force because the velocity is not changing. Drag Force Drag Force Drag Force Moment of release Before terminal After terminal Weight Force Weight Force Weight Force On the axes provided, draw a graph of the speed v of the coffee filter as a function of time t. Speed vs. Time T E A C H E R P A G E S
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. 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 air friction has no effect. vi 0 m/s g 9.8 m/s 2 y 1000 m v v 2g y 2 2 f i m v f 140 s b. If 26.7 m/s is approximately equal to 60 mph, what is the speed of the raindrop in miles per hour? 60 mph Speed in miles per hour = 140m s =315 mph 26.7 m s T E A C H E R P A G E S
Physics Not So Free Fall 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. Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 73
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 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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 2. 3. 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. Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 75
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 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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) 1 2 3 4 5 6 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 77
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 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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 3. 2. What does your graph indicate the exponent to be? Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 79
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 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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. Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 81
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 Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.
Physics Not So Free Fall 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? Copyright 2015 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org. 83