How Far and How High Does a Baseball Need To Go For an Out-of-the-Park Home Run?

The sportscasters for the Cleveland Indians knew they should be prepared for any eventuality when the new ballpark, Jacobs Field, opened in 1994. One possibility seemed to be that Albert Belle might hit a home run that went so high and so far that it left the ball park. What would they tell their listeners about the height and distance the ball went?

So they called in the Mission Possible team. The sportscasters wanted to know the distance from homeplate to the highest point of the stadium and the distance from the base of the outfield fence to the highest point of the stadium for every 5º from the 3rd base line around to the 1st base line. The problem is complicated by the fact that the distance from homeplate to the outfield fence varies from 325 feet to 410 feet. Furthermore, the height that would have to be cleared also varied, depending on where the ball was hit.

- First, how many sets of distances and heights do the sportscasters want?
- To find one set of solutions, start with the distance from homeplate across 2nd base to the outfield fence which is 410 feet at that point. Using a transit, you find that the angle of elevation from homeplate to the highest point that would have to be cleared for a homerun to go out of the park at that angle is 10º and the angle of elevation from the base of the outfield fence to that same point is 32.5º. Use this information to find the minimum distance from homeplate to the highest point of the stadium and the distance from the base of the outfield fence to the highest point of the stadium.
- Now you can tell the sportscasters that if they provide you with the two angles of elevation and the distance from homeplate to the outfield fence for each 5º movement around the stadium, you can provide them with all the statistics they need. After receiving the two angles of elevation for each 5º movement around the stadium, you decide that it would be faster to develop a general formula, so that by inputting the two angles and the distance from the homeplate to the fence, the distance from homeplate to the top of the roof and the height of the roof would be computed. Let A and B denote the angles of elevation to the top of the roof from homeplate and from the base of the outfield fence and let L be the distance from homeplate to the outfield fence. What are the correct formulas?
- Compare your formulas with those of other groups. Are they all the same? Are they all equivalent Which ones are simplest?
- Now write each correct formula in the form shown below:

$\displaystyle height = Lsin(A)sin(B)/sin(B - A)$

$\displaystyle distance = Lsin(B)/sin(B - A)$- What is the probable trajectory of a hit baseball on a windless day? Sketch. In what ways might the wind and other factors affect the path of the baseball?
- If a player actually did hit an out-of-the-park homerun, how would the distance the ball traveled compare to the figures you have been developing for the sportscasters? What other factors will add to the distance?
- Could there be a longest homerun? How would you measure it