Computational Model of Falling Sphere ONLY with Air Resistance Proportional to v2(1)

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City Colleges of Chicago, Harold Washington College *

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235

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Mathematics

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Apr 3, 2024

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docx

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Computational Model of Falling Sphere with Air Resistance Proportional to v 2 This set of exercises requires the student to generate a computational model of the 1D motion of a spherical object dropped from a tall building, and then graph and analyze the output of the model. It also guides the student in exploring the accuracy of a computational algorithm by comparing the computational results with an exact solution obtained analytically. The numerical approach used is the simple Euler method. Starting your spreadsheet program: If you have a Google account, you can open a Google sheet spreadsheet template If you don’t have a Google account or don’t want to use one, you can use Excel if you have it installed (it is free to CCC students), or you can try using the web-based version of Excel that should be available when you login to your CCC email (it should be available under the “App Launcher” in the upper left corner). Set up the “Control Panel” and define any variables and/or constants you will need for your spreadsheet program (give your variables and constants a meaningful name using the name box in Excel or Google Sheets). Choose reasonable and realistic values for any of the quantities that you need to include in your spreadsheet program. In the spreadsheet columns, insert the functions that will be need to define and that use the simple Euler method in order to calculate the values of the mass’s: • position. • velocity • acceleration Exercises Exercise 1: Computational Model of a Falling Sphere with Air resistance Produce a working computational model of a sphere that has been dropped from rest from a very tall building using the simple Euler method. Assume that the sphere will move entirely in one dimension, and that it is subject to the constant gravitational force near Earth's surface and to a resistive force proportional to the square of the sphere's instantaneous speed. Question 1 --At what time did the sphere reached terminal velocity? Highlight in yellow the spreadsheet boxes corresponding to your answer. Question 2 --How much time it takes a basketball to fall from The Willis Tower Skydeck (412 m) according to this model for air resistance? Highlight in yellow the spreadsheet boxes corresponding to your answer. Show your answers to the next two questions in a green high-lighted box close to your answer to Question 2 Question 3 -- How much time it takes a basketball to fall from The Willis Tower Skydeck (412 m) if air resistance is negligible?

Question 4 --How much larger is time of fall with air resistance? Upload your spreadsheet calculations and graphs in Brightspace assignment “Lab-Computational Model Motion_Falling sphere_air resistance” Exercise 2: Accuracy of Computational Model: Velocity vs. Time Since the computational approach is based on an approximation, it is important to determine just how small Δt should be for the approximation to accurately solve the 1D air resistance problem. Make a comparison between the time dependence of the velocity predicted by the computational model, and that predicted by the exact result, v y ( t )= √ 2 mg DϱA tanh ( √ DϱAg 2 m t ) . Use parameters that describe a 16-pound bowling ball (you should look up the diameter, and convert to meters), and let it fall a distance equivalent to the height of the Willis, tower (440 m). Assume the ball is initially at rest. Use a value of 0.5 for the drag coefficient, and the density of air near sea level. What value of Δt do you deem to be sufficiently small for the computational model to be accurate? Explain how you arrived at this value of Δt . Exercise 3: Accuracy of Computational Model: Position vs. Time Carry out the same comparison (computational vs. exact analytical solution) for the bowling ball's position as a function of time. The exact result for the ball's position is given by y ( t )= 2 m DϱA ln [ cosh ( √ DϱAg 2 m t ) ] . Assume the bowling ball is falling the same distance of 440 m. Do you find the same value of Δt , as found for the velocity comparison of Exercise 2, to be acceptable for the position comparison? Discuss this issue with other students and describe your conclusions here

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