This study is the result of an inquiry made by Matt Hagood of the Marlins' staff. Several people suspected that the estimated distances of the home runs hit at Pro Player Stadium were wrong. These are the distances the balls would have traveled if they had landed at field level instead of in the seats.
Matt uses a table which was produced around the time that the Marlins came into existence. The table has entries for every fifth row in every section of seats. For each location, there are three entries corresponding to three types of homers: line drives, average fly balls, and towering fly balls.
Locations for calculations
I have done calculations for 6 locations in the stadium. These include two locations along the left-field foul line (seating section 110-111), two locations along the right-field foul line (seating section 130-131), and two locations in the right field power alley (section 127-128). In each case, I made one calculation "at the rail" or right over the fence. In left field, I did a calculation for the first row of seats. I have assumed this is the row listed as "Row 25" in the tables. I also did row 25 for the other two sections.
The assumptions and techniques that went into doing the calculations are discussed below in some detail. I need to mention that the distances from home plate to each of my 6 locations and the vertical distance from the field level to these locations are based on measurements taken by Matt and myself in late April. We measured the distance to the left-field wall down the line to be 327.5 feet whereas the number on the wall is 330 feet. The distance to the right-field wall down the line is 347.25 feet but is labeled as 345 feet. And, the distance to the right-field power alley wall is 369.3 feet but is labeled as 385 feet. The vertical distances from field level to the top of the wall was measured to be 8 feet 2 inches. The vertical distance from the field to seats was estimated based on some measurements taken in the stands.
Technique and Assumptions
In doing these calculations I have included air resistance according to the prescription discussed in Robert K. Adair's excellent book The Physics of Baseball. This required some assumptions about the density of air in South Florida. This of course changes with the weather but should have little effect on the results reported here. The drag coefficient of a baseball is also required in the calculations. I used numbers from Adair's book. I have also assumed that the circumference of a baseball is 9.125 inches (rule book says the circumference must be no less 9.0 and no more than 9.25 inches) and that the weight of the ball is 5.125 ounces (rule book allows 5.0 to 5.25 ounces). I have assumed that the balls start out at 33 inches above the ground (waist high). I have also assumed that there is no wind.
My calculations produce distances of batted balls for various launch angles and initial speeds of the ball (as it comes off the bat). If you compare my results to those of Jim Carr at Florida State (see his Home run simulator page), you would see that my distances differ from his by about 1 to 10%. This pretty good considering the assumptions that went into the calculation. Some of the assumed parameters can easily be off by several percent. If I am off by as much as 5%, the distance that the ball travels for a fixed angle and speed will be off by about 2-3% (8 feet on a 400 foot homer).
However, if one requires a ball to land at a particular location in the seats, the projected distance to the ground from that location is probably not off by more than a couple of feet. To verify this, I have run my calculation with the drag force varied by 5% and found that the difference between distance traveled to change by only about 1.5 feet. So an even larger error in my initial assumptions would not have a significant impact on the results presented here.
The results of my calculations are summarized in the table and the three figures below. In doing the comparison to the Marlins' tables, I have made the assumption that a "line drive" corresponds to a ball launched at 30 degrees, an "average fly ball" corresponds to 35 degrees and a "towering fly ball" corresponds to 40 degrees. In my calculations have additional categories of "screaming line drive" and "Silo shot" corresponding to 25 and 45 degrees. I am not sure how good these assumptions about angles are.
In the table below the first number in each column represents the number from the Marlins' table and the second number is what I calculated. All distances are in feet.
|Location||Screaming line drive||Line drive||Average fly ball||Towering fly ball||Silo shot|
|Left field, down the line|
|Section 110-111 at the wall||-- 335||-- 333||-- 331||-- 330||-- 328|
|Section 110-111, row 25||-- 366||354 356||345 350||340 343||-- 338|
|Right field power alley|
|Section 127-128, at the wall||-- 376||377 373||374 372||372 370||-- --|
|Section 127-128, row 25||-- 465||457 459||448 454||443 449||-- --|
|Right field, down the line|
|Section 130-131, at the wall||-- 354||354 352||351 351||350 350||-- 347|
|Section 127-128, row 25||-- 444||438 438||428 433||424 428||-- --|
In the figures below I have made two types of graphs. The top graphs show the batted ball speed required for different launch angles for the ball to land in the location indicated. The bottom graphs show the distance for various launch angles. This is just the data from the table with a few more points thrown in. I have included colored bars on the bottom to indicate angle ranges for the five categories of homers (screaming line drive, line drive, etc.).
One thing that can be learned from the graphs is the estimated batted ball speed for the various types of homers. For example, a line drive hit just over the wall in the right-field power alley has a speed of about 103 miles per hour as it leaves the bat. One hit into the 25th row has a speed of about 120 miles per hour.
It would seem that the Marlins' tables are pretty good for short home runs but may be off by a few feet for longer homers. This may indicate that shots into the upper deck could be off by as much as 10-20 feet. A better estimate of distances to the locations may lead to better results. Checks for some of the upper deck numbers should be done. It would also seem that some of the concern about the tables isn't really due to the tables but due to the inaccurate distances posted at the walls. This should be fixed.
One unknown in my study is what angle constitutes a line drive or an average fly ball. Of course, using the Marlins' table requires a snap estimate of the angle anyway and is subject to judgment error...it is hard to estimate angles to better than 5 degrees.
If the current Marlins' tables are to remain in use, I recommend that a few more spot checks be made. If we decide the numbers are okay, we could supplement the tables with estimated batted-ball speed. This is a number that fans might find exciting since the long bombs require speeds in excess of 115 mph!