This Exercise Set has been submitted for peer review, but it has not yet been accepted for publication in the PICUP collection.

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What physics needs to be including in modeling a home run?

Developed by Fred Salsbury

This is a simplified version of a set of exercises I use in intermediate mechanics. The audience in mind are intro physics I students. The aim of this is for the students to figure out the importance of air resistance in modeling a typical home run in baseball. They will start from horizontal one-dimensional motion without air resistance or gravity up to modeling two-dimensional projectile motion with different models for air resistance.
Subject Area Mechanics First Year Spreadsheet See how Euler's method works (Exercise 1 for 1D, 4 for 2D) Observe Newton's first law (Exercise 1) Observe the effect or lack thereof of a linear model of air resistance on a small object in one-dimension. (Exercise 2) Observe the effect of a quadratic model of air resistance on a small object. (Exercise 3) Model simple projectile motion in two-dimensions (Exercise 4) Determine the effects of different models of air resistance on projectile motion (Exercise 5,6) Determine if any of these models are reasonable for a home room (Exercise 1-6) Assess the role of varying approximations in modeling particle motion (Exercise 1-6)
##How to model a home run in baseball? The overall goal in this exercise set is to go through a set of approximations to see what physics needs to be included in modeling a typical home run in baseball. In these exercises you will: Go from the simplest approximation, purely horizontal motion without gravity or air resistance (Exercise 1), then add in different approximations to air resistance (Exercises 2 and 3), then move to two-dimensional projectile motion (Exercise 4) without air resistance but with gravity, and then adding in different approximations to air resistance (Exercise 5 and 6). You should at the end of these exercises be able to 1) visualize the motion of a particle using graphs 2) verify Newton's first law 3) model air resistance 4) use Euler's method to calculate motion from different force laws 5) appreciate the role of changing approximations in modeling particle motion #####Parameters and initial conditions for these exercises are: A typical home run is hit at 49.2 m/s at an angle of 28.0 degrees. A baseball is typically hit at 0.760 m above the ground, and you might as well consider the initial x position is 0.000 A typical baseball masses 0.148 kg and has a diameter of 0.0740 m. The drag coefficients are 0.000197 for linear drag and 0.160 for quadratic drag. The magnitude of acceleration due to gravity is 9.81 m/s^2. Not all of these parameters will be used in each exercise, but deciding which parameters need to be used in each exercise is part of the problem. ###Exercise 1 The simplest approximation to model the motion of our baseball is to neglect both gravity and air resistance and consider the baseball motion as purely horizontal. Hopefully, you realize these are horrible approximations.Because this is the first exercise, the instructions will be the most detailed. **Question: Why are these approximations bad?** **Instructions** Start by opening the spreadsheet with the code template. Now we will fill out the spreadsheet to model the motion of our baseball/. 1) Fill in the missing parameters in the spreadsheet. 2) Now, for the physics, remembering that we are neglecting both gravity and air resistance, enter in the acceleration at time 0. 3) To figure out how to propagate the motion of our baseball, we will use Euler's method. Euler's method says that we will consider the acceleration as constant over some timestep, dt, which for these exercises are set at 0.1 secs, and we integrate by adding up little changes in velocity and position. _To do this in practice:_ A) click the cell corresponding to time = 0.1 secs for the velocity (D12 if the spreadsheet has not been modified), and make a formula that says that velocity is the previous velocity plus the previous acceleration times dt. If the spreadsheet has not been modified, this should read =D11+E11*dt . B) do the same for the position, where the formula is the previous position plus the previous velocity times dt. If the spreadsheet has not been modified, this should read =C11+D11*dt . C) Propagate the motion using Euler's method for the entire time-length specified. This means copy the formulas for acceleration, velocity and position down to cells 4035. The easiest way to do this is to copy (cntl-C or command-C) the formula in E11, then go to cell E12 look for the name box on the upper left of your screen, type E12:E4035 in that box to select those cells, then paste (control-V or command-V). Then repeat for the velocity formulas from D12 to D13:D4035, and the position formulas from C12 to C13:C4035. **Congratulations!** you have now propagated the motion of our baseball while neglecting gravity and air resistance, and approximating it as purely horizontal. Now you have to check your plots. 4) Check and see if the axes of the plots for position and velocity cover the entire range of your data. If not, click on the plots, and change the horizontal and/or vertical axis options so that the bounds include your entire data. 5) Now interpret your results: **A) Is the velocity plot consistent with Newton's first law? Why or why not?** **B) Is the position plot consistent with the kinematics you have seen in class? Explain** **C) Is this a good approximation to a home run? Explain.** ###Exercise 2: In the previous exercise, we had you make many serious approximations that made Exercise 1, a horrible approximation to a home run. There are several ways you can start to lift these approximations and include more physics. Let's start by lifting the approximation of no air resistance. The question then arises as to how to model air resistance. There are two common approximations, both of which involve a drag coefficient that depends on the object and the fluid and the cross-sectional area of the object and either the velocity of the object or the square of the velocity of the object. (Since you are in one-dimension, you don't have to worry about vectors.) In this Exercise, the force due to air resistance is modeled simply as $F_{d}=-DAv$, where D is the drag coefficent, v is the velocity, and A is the cross-sectional surface area. A is calculated automatically from the diameter in the spreadsheet. **Question: What are the units of the linear drag coefficient?** The procedure for this exercise follows closely on the previous exercise. 1) Enter in the parameters and initial conditions (NB: the area is auto-calculated via a formula,so don't overwrite). 2) Use your new force law for air resistance to calculate the acceleration at zero time (use A for the cross-sectional area, and don't forget to use Newton's second law to convert from force to acceleration). 3) Copy the formula for acceleration for all time steps (NB: the numbers won't be correct yet because you haven't done the velocity propagation, but the formulas will be.) 4) propagate the velocity and position using Euler's method as in Exercise 1. **Congratulations! ** you have now propagated the motion of our baseball while neglecting gravity and with linear air resistance, and approximating it as purely horizontal. Now you have to check your plot again and make sure the axes cover the range of your data. 5) Now interpret your results: **A) Is the velocity plot consistent with having drag?** **B) Is this a good approximation to a home run? Explain** ###Exercise 3) In Exercise 2, you saw what happened if you approximated the forces on a baseball by just linear drag without gravity and you assumed just horizontal motion. In this exercise, you will replace linear drag with quadratic drag. The physics here is almost the same as in Exercise 2. The only difference is that the force law depends on a different drag coefficient and the velocity dependence is quadratic: $F_{d}=-DAv{^2}$ . **Question:What are the dimensions of the drag coefficient for air resistance modeled as quadratic in velocity?** The procedure for this exercise is very similar to exercise 2. 1) include the appropriate parameters and initial conditions (there should only be one difference from Exercise 2) 2) calculate the acceleration using a quadratic force law for air resistance for different timesteps 3) propagate velocity and position using Euler's method 4) make sure your plot axes include all the data. 5) Now interpret your results: **A) Is the velocity plot consistent with having air resistance?** **B) Compare the effects of quadratic and linear approximations to air resistance.** **C) Is this a good approximation to a home run? Explain** **D) Is it better or worse than linear approximation to air resistance. Explain.** ###Exercise 4) In the previous three exercises, you modeled a baseball's motion as purely horizontal and neglected the effect of gravity. Despite these serious approximations, you hopefully gained an appreciation of how easy it is to propagate a particle's motion given a force law, as well how different models of air resistance affected the motion of your baseball in one-dimension. In this Exercise, you will improve your model drastically by considering a home run as two-dimensional under the influence of gravity, but without air resistance. In exercises 5 and 6, you will combine modeling of gravity with air resistance; you might be able to hypothesize already which model will be the most accurate. The procedure is a slight generalization of what is found in Exercise 1. 1) include the appropriate parameters and initial conditions. Remember that there are initial positions now in both x and y. Also remember that 49.2 m/s is the speed of the baseball. You will need to convert to x and y components. Also remember that you will need to convert 28 degrees to radians 2) determine the acceleration in both x and y directions and propagate to all timesteps Don't over think this 3) propagate velocities and positions using Euler's method Remember there are two velocities and two positions, however, using Euler's method you propagate $v_{x}$ using $a_{x}$, $v_{y}$ using $a_{y}$, x using $v_{x}$ and y using $v_{y}$. 4) make sure your plot axes include all the data. 5) Now interpret your first results: **A) Note that y became very large and negative. What does this mean? Since this is not physically reasonable, what force have you neglected?** 6) Since y<0 is physically unrealistic, change your axes so that your x and y plots (height and range) only include times where y is >0. 7) Now that you are only including physically realistic results **B) Have you improved your modeling of a home run?** **C) What is the percent error? Don't use an absurd number of significant digits.** ###Exercise 5 In the last exercise, you modeled a home run as two-dimensional projectile motion without air resistance. In this exercise, you will add in air resistance modeled as linear in the velocity. 1) add the appropriate initial conditions and parameters, most will be the same as in Exercise 4. 2) modify the acceleration from exercise 4 to include air resistance that is linear in velocity for each timestep. $\vec{F_{d}}=-DA\vec{v}$ Since this is multi-dimensional motion, the velocity, $\vec{v}$ is a vector and must be divided into x and y components. 3) propagate velocities and positions using Euler's method 4) Since y<0 is physically unrealistic, change your axes so that your x and y plots (height and range) only include times where y is >0. 5) Now that you are only including physically realistic results **A) Does modeling air resistance as linear in the velocity change the results significantly from neglecting air resistance?** **B) What is the percent error? Don't use an absurd number of significant digits**. ###Exercise 6 You will now in this final exercise, model air resistance as quadratic in the velocity, similar to what you did in Exercise 3. However, since you are modeling a multi-dimensional system, you need to consider vectors and must be careful. A quadratic approximation to air resistance has the following form: $\vec{F_{d}}=-DAv{^2}\hat{v}$ . However, $v^{2}\hat{v} =v^{2}*\frac{(v_{x} \hat{x}+ v_{y}\hat{y})}{(v)}=v\(v_{x} \hat{x}+ v_{y}\hat{y})$. So that the force needs to include the speed v, but we can indeed break up the force into x and y componenets. 1) add the appropriate initial conditions and parameters, most will be the same as in Exercise 5. 2) modify the acceleration from exercise 5 to include drag that is quadratic in velocity for each timestep, recalling the discussion above about how to calculate the force due to the quadratic approximation to air resistance. 3) propagate velocities and positions using Euler's method 4) Make sure to also calculate the speed at each timestep so that the acceleration can be calculated correctly. 5) Since y<0 is physically unrealistic, change your axes so that your x and y plots (height and range) only include times where y is >0. 6) Now that you are only including physically realistic results **A) Does modeling air resistance as quadratic in the velocity change the results significantly from neglecting air resistance?** **B) What is the percent error? Don't use an absurd number of significant digits.**