13.7: Exercises

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Before you go on, you might want to work on the following exercises.

Exercise 13.1

Manny Ramirez is a former member of the Boston Red Sox who was famous for his relaxed attitude. The goal of this exercise is to solve the following Manny-inspired problem:

What is the minimum effort required to hit a home run in Fenway Park?

Fenway Park is a baseball stadium in Boston, Massachusetts. One of its most famous features is the “Green Monster,” which is a wall in left field that is unusually close to home plate, only 310 feet away. To compensate for the short distance, the wall is unusually high, at 37 feet.

You can solve this problem in two steps:

For a given velocity, find the launch angle that maximizes the height of the ball when it reaches the wall. Notice that this is not quite the same as the angle that maximizes the distance the ball travels.
Find the minimal velocity that clears the wall, given that it has the optimal launch angle. Hint: this is actually a root-finding problem, not an optimization problem.

Exercise 13.2

A golf ball hit with backspin generates lift, which might increase the distance it travels, but the energy that goes into generating spin probably comes at the cost of lower initial velocity.

Write a simulation of the flight of a golf ball and use it to find the launch angle and allocation of spin and initial velocity (for a fixed energy budget) that maximizes the horizontal range of the ball in the air.

The lift of a spinning ball is due to the Magnus force (see https://greenteapress.com/matlab/magnus), which is perpendicular to the axis of spin and the path of flight. The coefficient of lift is proportional to the spin rate; for a ball spinning at 3000 rpm it is about 0.1. The coefficient of drag of a golf ball is about 0.2 as long as the ball is moving faster than 20 m/s.