Mini-project 3 Tennis ball launcher Mini-Project 3 requires you to use MATLAB to model the trajectory of a tennis ball being shot from a tennis ball launcher to a player. The tennis ball trajectory model that you create for Mini- Project 3 will build on the models for projectile motion that you have created. You will use the model you create for the trajectory of the tennis ball to explore how the launch angle, initial velocity, and distance of the launcher from the baseline affect flight time and ability to hit a target (in this case, the player s racket). 1 Model for the Trajectory of a Tennis Ball The coordinate system that you will use for this project will have an origin that is located on the ground at the center of the baseline on the launcher s side of the net. The positive x-direction of the coordinate system is pointed horizontally toward the net, and the positive y-direction is pointed up in the vertical direction, as shown in Figure 1. Using this coordinate system and the assumption that the tennis ball stays in the x-y plane, a tennis ball moving through the air can be modeled by the equations of motion shown in Equations (1) through (6). The relevant parameters for modeling the trajectory of a tennis ball are given in Table 1. Figure 1. Coordinate system for launcher problem d d 1 2, (1) 1
d d 1 2, (2) d d, (3) d d, (4) 0 cos, and (5) 0 sin, (6) Table 1. Parameters for modeling a tennis ball launch Parameter Description Value with units Mass of tennis ball 0.0585 kg Gravitational acceleration 9.81 m/s 2 Radius of tennis ball 0.0343 m Density of air 1.23 kg/m 3 Drag coefficient 0.54 Projected cross-sectional area of the tennis ball To improve your ability to hit the target, you will vary the following three parameters: the initial velocity of the ball,, the launch angle,, and the distance of the launcher from the baseline, d. The initial position, 0 and 0, is the release point of the tennis ball from the launcher. Note that both the x-coordinate and the y-coordinate of the initial position will be dependent on and d as well as the height of the launcher base,, and the length of the launcher arm,. The relevant parameters for the launcher, along with the initial position 0 and 0, are shown in 2
Figure 2, and the measurement of is shown in Figure 1. Relevant parameters for the launcher and tennis court dimensions are provided in Table 2. Equations (7) and (8) show how to calculate the initial position based on the given values and and the chosen values and. Figure 2. Tennis ball launcher with dimensions 0 (7) 0 (8) Table 2. Launcher and Tennis Court Dimensions Parameter H L Length of tennis court (baseline to baseline) Net position from baseline Net height Value with units 0.3 m 0.3 m 23.75 m 11.875 m 1.07 m 3
Your goal for this project is to determine a combination of initial velocity, launch angle and position from the baseline such that the ball lands as close as possible to a specified target in the shortest amount of time. There are several constraints for the launch and trajectory that will affect your analysis: 1. The ball must be launched such that it is able to clear the net (net height is given in Table 2). 2. The ball must bounce once on the player s side of the net before it reaches the player (note: the player is on the opposite side of the net from the launcher). During the bounce, some of the ball s kinetic energy is converted to other forms of energy. As a result, the ball s velocity after the bounce will decrease. This reduction in kinetic energy is represented by the coefficient of restitution, which affects the vertical component of the velocity as shown in Equation (9), where is the coefficient of restitution (0 1), is the vertical velocity before the impact, and is the vertical velocity after impact. Note that Equation (9) includes a negative sign since the direction of the vertical velocity reverses after the bounce. The horizontal component of the velocity is unaffected. (9) 3. For physical reasons, the following limitations will apply to the parameters you vary: The launch angle is limited to 0 90, where 0 launches the ball horizontally toward the net, and 90 launches the ball straight upward. The distance of the launcher from the baseline is limited to 0 m 11 m. The launch velocity is limited to 1 35. The x-coordinate for the target point is limited to 12 m 29 m. 4
2 Determining the Optimal Trajectory Minimizing the ball s time of flight and hitting as close to the target as possible are conflicting goals. To determine when you have achieved the best compromise between these two goals, you will need to minimize a performance index that considers the ball s flight time and how far away from the target the ball hits. The performance index PI that you will use is shown in Equation (10). PI 10 (10) In Equation (10), represents the ball s time of flight, and is the distance between the ball s launch location and the target player location, shown in Equation (11), where and denote the desired target. The term represents the minimum distance between the ball s actual trajectory and the target, shown in Equation (12). 0 0 (11) min (12) The trajectory whose initial velocity, launch angles, and start position yield a minimum value for the performance index PI in Equation (10) is considered the optimal ball trajectory. 3 The test exercise During class on Thursday of Week 10 (May 17, 2018), we will hold an exercise to test the results of your program. The test exercise procedure is as follows: 1. A prediction card will be passed out to each team. 2. I will provide you with the ball s test-day target location, and, and coefficient of restitution,. Teams will have 20 minutes to complete Step 3. After that, no more submissions will be accepted. 5
3. Teams will use their MATLAB code to determine the ball s initial velocity, launch angle, and distance of the launcher from the baseline that yield the optimal trajectory according to the performance index in Equation (10), as well as the corresponding performance index value, the predicted final position of the ball, and the predicted time of flight. Teams will record these results on their prediction cards. Program results must be left on the teams computer screens. 4. I will verify that the reported values are the ones appearing on the teams computer screens. 5. The optimal initial velocity, launch angle, and distance from the baseline predicted by each team will be tested in a virtual experiment (i.e., a computer simulation). 4 Submitting your MATLAB code By 5 pm on Friday of Week 10 (May 18, 2018), upload your team s well-commented MATLAB code (main routine and functions) as m-files (.m) to Moodle. Upload only ONE submission per team. If your code failed to work at the test exercise, fix the code before you upload it to Moodle. 5 Project grading Your grade for this project will be computed with the following weights: 40 pts Test exercise prediction 60 pts MATLAB code (well-documented and easy to follow) 100 pts Your test exercise prediction will be scored based on the actual value of the performance index PI associated with your submitted launch conditions: the closer you are to my minimum value of PI, the higher your score. Your test exercise prediction score will be determined as follows, where Figure 5 illustrates the scoring function: Percent difference your actual PI value my minimum PI value my minimum PI value 6 100%
0 for no valid submission 10 for a solution not meeting constraints score 24 for 50% 40 0.32 for 0% 50% 40 38 Test exercise performance score (pts) 36 34 32 30 28 26 24 Figure 5. Illustration of the scoring function for your test exercise prediction. 6 Suggested steps This is a big project, so I suggest the following steps for you to use in breaking the project down into more manageable pieces. You are not required to follow these steps, but you may find them helpful. If you do take my advice, you are free to work faster than what is suggested, but to keep from falling too far behind, you should try to keep up with this schedule. Use the values listed in Tables 1 and 2 as needed. Day 1 (Friday): 22 20 0 10 20 30 40 50 60 70 80 90 100 Percent difference, p 1. Create and test a function to compute the ball s trajectory with no air drag. For now, don t worry about making sure the ball bounces just calculate the trajectory until the 7
ball hits the ground. Write a main program that calls the ball trajectory function. Run your code and make a plot of the ball s trajectory. Because there is no drag, the ball s trajectory in the plane must be parabolic. Be sure to try several different values for, and. If you use 10,3 m,θ40, you will get a plot like Figure 3 below. 2. Modify your ball trajectory function to include the effect of air drag. Run your code to verify that, because of drag, the ball s trajectory is no longer parabolic in the plane and does not travel as far. Figure 3 shows the trajectory with and without drag for the input parameters listed in Step 1. The corresponding landing location and flight time for a time step of 0.0001 s are as follows (note that since we re running the code until the ball hits the ground, 0 m): Without drag: 13.8 m in 1.4 s With drag: 12.4 m in 1.3 s Figure 3. Tennis ball launch with and without drag Day 2 (Monday): 8
1. Modify your ball trajectory function again, this time adding the bounce (drag should still be included). An important consideration for this portion of the project is that when the bounce occurs, the velocity along the y-axis changes direction, and the ball begins to move upward. This means you will have to separate Euler s method into two stages one for before the bounce and one for after the bounce so that you can account for the change in direction. Here is a sample result for you to check against (Figure 4 shows the plotted solution), where the code runs until the ball hits the ground after the bounce: For 10 m/s, 40, 3 m, 0.5, and 0.0001 s: the ball lands at 16.4 m in 2 s. Figure 4. Tennis ball launch with bounce 2. Now, consider how you might check that your solution meets the problem constraints. How can you check to see if the ball makes it over the net? How can you check to see that the ball lands on the player s side of the net? You may also want to think about how 9
changing affects the ball s trajectory? We strongly encourage you to include the net and the target on your trajectory plot to help you see if your results make sense. 3. Incorporate the performance index in your program. Here is a sample result you can use to check your calculation: For 9 m/s, 52, 6 m, 0.7, 0.0001 s, and a target of 15 m and 1 m, PI 1.92. Day 3 (Tuesday): You may have noticed that when is small for accuracy, it takes a little while to compute even one trajectory. It could take a very long time to run through 100 different combinations of launch conditions, never mind 1000 or so, and thus we will definitely want to speed things up. We can do this by pre-allocating the solution vectors. In your code, it would not be surprising if you had something like the following structure, which is code for a ball dropping under the influence of gravity: clc clear variables g = 9.81; dt = 0.0001; maxsteps = 1e7; v(1) = 0; x(1) = 0; for i = 1:maxsteps v(i+1) = v(i) - g*dt; x(i+1) = x(i) + v(i)*dt; end When we start the for loop, the vectors v and x are only one element long, and each time we go through the loop the vectors get longer by one element. The process of making the array longer is very time consuming in MATLAB, especially when the vectors are large. This little program takes about 5 seconds to run on my machine (yours may be faster or slower). If we could tell MATLAB how big the vectors were going to get, it could make them that big to begin with and the for loop would run much more quickly. 10
Now consider the code shown below: clc clear variables g = 9.81; dt = 0.0001; maxsteps = 1e7; v = zeros(1,maxsteps+1); % Makes a vector v with (maxsteps + 1) % zero elements x = zeros(1,maxsteps+1); % Makes a vector x with (maxsteps + 1) % zero elements v(1) = 0; x(1) = 0; % Not actually necessary now because v(1) = 0 already % Not actually necessary now because x(1) = 0 already for i = 1:maxsteps v(i+1) = v(i) - g*dt; x(i+1) = x(i) + v(i)*dt; end The two new lines are the pre-allocation of the vectors v and x; we set them up and then execute the for loop. Running this now takes about 0.2 seconds! I strongly encourage you to consider pre-allocating vectors for your tennis ball trajectory code. The only drawback to this method is that you must estimate the length of the vectors you will need. If you guess too small, depending on the structure of your code, your program will either slow down again or the ball will not have hit the ground by the end of integration. If you guess too large, you will get an Out of Memory error there is not enough memory for you to use! Use one of your previous runs and the time step you have selected to estimate how long the vectors need to be. Once you have implemented vector pre-allocation in your program, think of a way to determine the optimal combination of the initial velocity, launch angle, and distance of the launcher from the baseline. Of course, you may use a guess-and-check approach, but if you could automate some or all of this process, then your search will likely be much faster. This could be useful on the day of the test exercise when you have only 20 minutes to determine a good set of launch conditions. Keep in mind that there are many, many possible combinations of these three parameters let MATLAB find a good combination for you! 11