DAD Lab 10 Air Resistance

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Apr 3, 2024

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DAD Lab 10: Air Resistance [Do-At-Domicile] Team members: __Nora Peterson, Caitlyn Miller 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 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 describe it with the term terminal velocity, or v T . 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 F drag = –bv (for viscous drag) or F drag = –cv 2 (for inertial drag). The constants b and c are called the drag coefficients that depend on the size and shape of the object among other things. 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 ∝ m or v T 2 ∝ m 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. 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. 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? What kind of mathematical relationship do you predict will exist between the velocity of fall and the number of filters? Adapted from a TI and Vernier lab by Larry Browning

The fall was faster when the two filters were together. This indicates that weight correlates to velocity. This is also shown in v = √ 2 gh 2. If there were no air resistance, how would the rate of fall of a coffee filter compare to the rate of fall of a baseball? If there were no air resistance, both would fall at the exact same rate. 3. Sketch your prediction of 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? At terminal velocity, there is no more force of acceleration, and the net force would be zero. DATA COLLECTION Use Video Analysis to determine the terminal velocity of one or several coffee filters as they fall. To do this, in Logger Pro go to File/Open/Experiments/Sample Movies/Coffee Filters as indicated here: Adapted from a TI and Vernier lab by Larry Browning

This is a composite video of six stacks of coffee filters falling in air. Digitize the fall of each of the six coffee filters. Your data should indicate a leveling off of the velocity vs. time plot as the filters reach terminal velocity and indicated in Figure 2: Velocity vs. Time – Single Filter. When the filters have achieved terminal velocity, the Y vs. t data should be linear and the Y-Velocity vs. t data should be constant as mentioned above. In the plot below you can see both the Y vs. t and the Y- Velocity vs t graphs for a single coffee filter. Both indicate a terminal velocity of about - 0.9 m/s. You may use either of these methods to determine the terminal velocity ( V T ) to complete the following table. Keep in mind, there is the possibility that the filters may not reach V T before they reach the ground. Trial # of Filters Terminal Velocity ( V T ) (Terminal Velocity) 2 or ( V T 2 ) 1 1 -.8304 m/s .6896 m 2 / s 2 2 2 -1.254 m/s 1.573 m 2 / s 2 3 3 -1.5 m/s 2.25 m 2 / s 2 Adapted from a TI and Vernier lab by Larry Browning

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4 4 -1.706 m/s 2.91 m 2 / s 2 5 5 -2.001 4.004 m 2 / s 2 6 6 -2.128 4.528 m 2 / s 2 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. This can be easily done with Logger Pro. 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 V T ∝ m. Or, if the drag force is proportional to the square of velocity, then V T 2 ∝ 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? The velocity squared graph has a straighter 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 (viscous or inertial drag) —one or both may not correspond to what you observed. The -bv models the data better because it’s correlation is close to 1; -cv^2 however goes through the origin like the graph we chose above. 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? As more filters are added, the time it takes for them to fall will decrease. If one filter takes 1t, four filters will take ¼(t). Adapted from a TI and Vernier lab by Larry Browning

EXTENSION 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, ΣF = ma = 0, must also be zero 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.) ∑F = −mg + bv T = 0 or ∑F = −mg + cv T 2 = 0 Adapted from a TI and Vernier lab by Larry Browning

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