1. Develop a matlab program that uses either second or fourth order Runge-Kutta method to determine the trajectory of the ball, given initial launch angles (phi, theta). You will want to treat the motion in each direction separately (though the drag force depends on the magnitude of the velocity). You might want to consider solving the problem in stages: for example, you might want to first use the Euler method without drag or lift, then add drag and lift, then change it to use second order RK, etc. Your program should determine whether the ball went through the goal area before hitting the ground (i.e. was a good goal). Your final code should use the RK2 or RK4 method, and treat the drag and lift forces properly, and allow for any pair of launch angles (phi, theta). The code should output the final result: Did the kick result in a goal, did it miss, or did it hit the ground before crossing x = 0? Be careful to consider the radius of the ball when making these determinations. Figure 4. Stem3 plot of a goal-making ball trajectory. Units are meters. Starting speed is 30 m/s and spin is 10 rotations/s around the z axis.

2. Find at least 4 combinations of horizontal and vertical launch angles (phi, theta) that result in the ball trajectory going into the goal before the ball hits the ground (i.e. without bouncing). Your successful angle values should be printed to the command screen so the user can see them. Feel free to work with integer values for the angles, in degrees. (So, try theta = 45, 46, 47, etc. degrees. Hint: you can solve this by trial and error, but it might take a long time to find 4 successful shots; you could use for loops to check lots of possibilities in one run.)

3. Pick one successful combination of kick angles and do the following:

A. Make a three dimensional plot of the balls trajectory. You can use the mouse to rotate the plot on the screen, which provides good 3D perspective. Or, you can use the view (theta, phi) command to change the orientation of the plot (this also works well for plotting to a file). You might consider using Matlabs stem3() command. It plots each point with a stem, i.e., a line going from (x, y, 0) to (x, y, z).

B. Answer this question: what would the path of the ball be if there were no air drag and no lift force?

C. Imagine a friend of yours who has taken Intro Physics, but not 281, takes a peek at your trajectory from part (A) above. They say: wow, why does the balls path bend like that??. Write a short (a few sentences) answer that would be simple enough for your friend to understand, but also detailed enough to convey the most important ideas.