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Falling Sphere with Air Resistance Proportional to $v^2$
Developed by Kelly Roos  Published July 17, 2016
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.
Subject Area  Mechanics 

Levels  First Year and Beyond the First Year 
Available Implementations  C/C++, Fortran, Glowscript, IPython/Jupyter Notebook, Mathematica, MATLAB, Python, Spreadsheet, and Haskell 
Learning Objectives 
Students who complete these exercises will be able to:
 model the motion of a falling sphere with air resistance in one dimension using the Euler algorithm (**Exercise 1**);
 produce graphs (position and velocity vs. time) of the computational solution (**Exercises 26**);
 assess the accuracy of the computational solution by comparing it to the analytical solution (**Exercises 2 and 3**);
 describe changes in the behavior of the model (e.g., time to approach terminal velocity) based on changes to properties of the falling sphere (e.g., mass and crosssectional area) (**Exercises 46**);
 describe ways to test the accuracy of a computational solution when there does not exist a known analytical solution (**Exercise 7**).

These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
#### 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.
#### 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 $\Delta t$ 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,
$$v_{y}(t)=\sqrt{\frac{2mg}{D\varrho A}}\tanh\left(\sqrt{\frac{D\varrho Ag}{2m}}t\right).$$
Use parameters that describe a 16pound bowling ball (you should look up the diameter, and convert to meters), and let it fall a distance equivalent to the height of the Sears, oops  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 $\Delta t$ do you deem to be sufficiently small for the computational model to be accurate? Explain how you arrived at this value of $\Delta t$ .
#### 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
$$y(t)=\frac{2m}{D\varrho A}\ln\left[\cosh\left(\sqrt{\frac{D\varrho Ag}{2m}}t\right)\right].$$
Assume the bowling ball is falling the same distance of 440 m. Do you find the same value of $\Delta t$, as found for the velocity comparison of Exercise 2, to be acceptable for the position comparison?
#### Exercise 4: Position and Velocity of Dropped Bowling Ball
Produce plots of the bowling ball's velocity and vertical position as functions of time from the results of the computational model, using the parameters from the previous exercises and the value of $\Delta t$ (determined in Exercises 2 and 3) that produces a tolerably accurate computational solution. Has the bowling ball reached its terminal velocity by the time it hits the ground? Use your model to predict the time required for the bowling ball to fall the full 440 meters to the ground.
#### Exercise 5: Position and Velocity of Dropped Basketball
Repeat Exercise 3 using the parameters for a basketball in your model (you will have to be resourceful and look up the weight and diameter of a typical basketball). How does the behavior of the basketball compare to that of the bowling ball?
#### Exercise 6: Position and Velocity of Dropped Baseball
Repeat Exercise 3 using the parameters for a baseball (again, look up the relevant parameters). How does the behavior of the baseball compare to that of the bowling ball and basketball?
#### Exercise 7: Assessing Computational Accuracy in the Absence of an Analytical Solution
In practice, computational models are produced to describe the dynamics of systems for which an exact analytical solution is very likely unknown. Based on your observations of the behavior of the model as $\Delta t$ gets smaller, can you suggest a procedure that could be used to assess the accuracy of a computational model in the event that you have no analytical curve with which to compare the computational result? Be very detailed in describing the procedure.
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Credits and Licensing
Kelly Roos, "Falling Sphere with Air Resistance Proportional to $v^2$," Published in the PICUP Collection, July 2016.
The instructor materials are ©2016 Kelly Roos.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license