#### The Throwing a Baseball Problem(1)

A map of a portion of a college campus is given in Figure 1. The map shows the walking paths and dormitories in this section of campus and the approximate distances (in 100 feet) between locations. Your roommate has convinced you to open a hot dog stand on weekends at one of the intersections along the walkways. You would like the stand to be as convenient as possible for the students.

Where on campus should you set up your stand?

How does your location change if you set up two stands?

Suppose A and C are female dorms and D, E, and F are male dorms. How would your location change if 30% of females and 80% of males are likely to eat at your stand.

Suppose the path between B and C and the path between E and D go uphill and that it is twice as hard to walk uphill as downhill. How would your choice change?

#### The Throwing a Baseball Problem(2)

You are in Fenway Park in the Summer of 1997. The Red Sox are playing host to the Seattle Mariners. The Mariner's ace randy Johnson, a.k.a. "The Big Unit", is on the mound in the top of the first, facing Big Mo Vaughn of the Sox. It seems that Johnson, known to be somewhat wild, has walked the bases loaded on just 12 pitches. The Mariner's manager, "Sweet Lou" Pinella, has just come out to the mound.

"Hey Unit, looks like you have pitched yourself into quite a jam."

"Yeah coach, I am having trouble with my release point. Every ball I throw is high. My release point must be all messed up. Did you guys see anything from the dugout?"

"You bet, our theory is that the force vector for your pitch is at an improper angle, although its magnitude appears to be correct."

"Wow!! A vector problem, we're going to need a yearling."

Suddenly, out of nowhere, a yearling appears looking positively stunning in his white over gray and yellow brass. "Never fear , I'm in MA205!!!"

The yearling, the Big Unit, and Sweet Lou begin to examine the problem by making a drawing in the dirt of the pitcher's mound. Here is their drawing:

The drawing shows that the Big Unit releases the ball at a height of 11 feet 8 inches above home plate from a horizontal distance of 60 feet 6 inches from home plate. He releases the ball with an initial velocity, $V_0$, at a positive angle alpha below horizontal. In order to be a strike, the ball must end up below Mo Vaughn's belt buckle and above his knees. Mo's knees are one foot above the ground. His belt buckle is three feet above the ground.

1. Find the range of values for the angle alpha which will result in the "Big Unit" throwing a strike.

2. Choose an angle alpha which will force the ball to cross the plate at an elevation of three feet. Find the speed of the ball when it crosses home plate, the time it takes for the pitch to reach home plate, and the arc length for the pitch. Determine the components of the velocity vector as the ball crosses the plate.

3. If the Big Unit throws a waist high strike, Mo will hit the ball at an angle, gamma, above horizontal. The initial velocity of this batted ball is $H_0$. The left field wall in Fenway Park is 318 feet from home plate and is 38 feet high. If the ball goes over the wall, it will be a home run. Assuming randy Johnson throws a strike, will Big Mo hit a home run?

4. If Big Mo does not hit the ball over the wall, the left fielder will try to catch the ball. Place the coordinate axes at the center of home plate with the $y$-axis on the third base line, the $x$-axis on the first base line, and the $z$-axis straight up and down through home plate. Measuring position in units of feet, the left fielders glove (attached to his hand) is at coordinates $(30,200,5)$. NOTE: We will assume that the glove stays at a constant elevation of 5 feet above the ground throughout the play. The initial position of the baseball is $(0,0,3)$. In other words, the ball is three feet directly above home plate when it is struck. Assume the ball is hit in a direction? degrees from the 3rd base line. When the ball is hit, the left fielder will move in a straight line directly to the position where he will catch the ball five feet above the ground. The direction the left fielder takes to get to the ball can be expressed as an angle beta measured from the relationship of his direction to the left field line. (See sketch). Find the angle Beta. Determine the minimum speed at which the left fielder must travel at this angle Beta to get to the ball before it drops below 5 feet. (We don't allow diving catches). Assume the maximum speed of the left fielder is 20 miles per hour. Will the left fielder catch the ball?

5. Make a 3 dimensional plot of Mo's hit.

Specifications:

Velocity of the pitch, Initial velocity of the batted ball, Angle of elevation of the batted ball, Angle from third base line of batted ball