The Physics of Baseball
Alan M. Nathan
University of Illinois

With the 2007 World Series being played between the Boston Red Sox and the Colorado Rockies, the question naturally arises regarding the affect of the mile-high altitude at Coors Field in Denver on the trajectory of a baseball in flight compared to a trajectory at Fenway Park in Boston. I address this question in this article. In addition, I discuss the effect of the famous Coors Field humidor. At that time, I did an interview with the Boston Globe, resulting in the article entitled "In the thick of the series, thin air may toss Sox a curve", by Colin Nickerson, which appeared in the Saturday October 27, 2007 edition.

Forces on a Baseball in Flight

Regardless of whether the ball is pitched or batted, its motion is determined by the forces acting on it. These forces are the downward force of gravity that we are all familiar with as well as two principal aerodynamic forces: the drag force and the Magnus force. Both the drag and Magnus forces result from small imbalances of the air pressure on different parts of the ball. For a ball at rest in the air, pressure is the result of air molecules randomly bouncing off the surface of the ball. This type of pressure is known as static pressure, since the ball is not moving relative to the air. The static pressure on various parts of the ball is equal so that there is no net force on the ball pushing the ball one way or another. That is, the force due to air pressure on one side of the ball that tries to push the ball one way is exactly balanced by the air pressure on the other side of the ball pushing it the opposite way. When the baseball is moving through the air, it experiences what is known as dynamic pressure. Looked at from the point of view of the ball, the on-coming stream of air molecules collides with the surface of the ball, pushing the ball backwards. Said differently, there is a net force on the ball that is exactly opposite to its direction of motion. This force is call the drag force, although it is also commonly referred to as "air resistance". The drag plays an extremely important role in the flight of a fly ball. For example, a fly ball that carries 400 ft would carry about 700 ft if there were no drag. The drag plays a less significant -- but still important -- role in the flight of a pitched baseball. Roughly speaking, a baseball loses about 10% of its speed during the flight between pitcher and catcher, so that a baseball that leaves the pitcher's hand at 95 mph will cross the plate at about 86 mph.

If the baseball is also spinning, it experiences the Magnus force, which is responsible for the curve or "break" of the baseball. The direction of the force is such that the ball breaks in the direction that the leading edge of the ball is turning. For example, a baseball thrown with backspin (e.g., an overhand fastball) has an upward Magnus force, opposing gravity, so that a typical fastball does not drop as much as it would if it were solely under the influence of gravity. On the other hand, an overhand curveball ("12-6") has topspin, so that the Magnus force is down, in the same direction as gravity. Such a pitch will drop more than it would from gravity alone. Other pitches (sliders, cutters, etc.) have a sideways component of spin, leading to a side-to-side break of the pitch. The Magnus force also has an important effect on the flight of batted baseballs. For a fly ball on a typical home run trajectory, the ball usually backspin. The upward Magnus force opposes gravity, keeps the ball in the air longer, and leads to a longer fly ball.

The important thing to remember about both the drag and Magnus forces is that they both are proportional to the density of the air. If baseball were played in a vacuum with no air, these forces would be exactly zero. But baseball is not played in a vacuum and there are wide variations in air density at different altitudes. Higher altitudes mean lower air density. Higher temperatures also mean lower air density. See Air Density Calculator to calculate the air density at a given altitude and temperature. When you do so, you will find that the air density in Denver (5280 ft) is about 82% of that at sea level (0.0627 vs. 0.0764 lb/cubic ft at 70 deg F). Therefore, we can expect that the drag and Magnus forces in Coors will be about 82% of their values at Fenway.

By the way, another thing that can affect the density of air is the relative humidity. It is often thought that humid air is more dense than dry air. In fact, it is just the opposite: humid air is less dense than dry air for the simple reason that a water molecule weighs less than an air molecule. See this link for a nice discussion.

UPDATE (July 14, 2008) I found a web site that gives a calculation of air density as a function of altitude, air temperature, altimeter setting, and relative humidity. The altimeter is related to, but not the same as, the elevation-corrected air pressure. See this link for a handy tool. This site says that the ICAO International Standard Atmosphere standard conditions for standard sea level air density are 0 meters (0 feet) altitude, 15 deg C (59 deg F) air temp, 1013.25 mb (29.921 in Hg) pressure and 0 % relative humidity ( absolute zero dew point). The standard sea level air density is 1.225 kg/m^3 (0.00764 lb/ft^3).

Effect of Altitude on Pitched Baseballs

In light of the above discussion, we can expect two main effects. First, pitched baseballs will be a little bit quicker in Coors than at Fenway. Since a baseball loses 10% of its speed in Fenway, the average speed is about 95% of its peak speed. At Coors, the baseball loses only 8% of its speed, so that the average speed is about 96% of its peak speed, or about 1 mph faster. This effect is quite small and batters can easily adjust to it. Far more important is the reduction in the Magnus force, resulting is less break on the pitched ball. If we take 18 inches as a typical break at Fenway due to the Magnus force, an identically thrown pitch will break only about 82% as much--or about 14-15 inches--at Coors. For example, an overhand curveball will drop about 4 inches less at Coors. An overhand fastball will drop about 4 inches more at Coors. Why "more"? Because the upward Magnus force opposing gravity is less at Coors. A ball thrown with pure sidespin will have a sideways break about 4 inches less at Coors. Generally speaking, less break or movement favors the batter.

Effect of Altitude on Batted Baseballs

The reduced drag and Magnus forces at Coors will have opposite effects fly balls on a typical home run trajectory. The principal effect is the reduced drag, which results in longer fly balls. A secondary effect is the reduced Magnus force. Remember that the upward Magnus force on a ball hit with backspin keeps it in the air longer so that it travels farther. Reducing the Magnus force therefore reduces the distance. However, when all is said and done, the reduced drag wins out over the reduced Magnus force, so that fly balls typically travel about 5% farther at Coors than at Fenway, all other things equal [1]. Therefore a 380 ft drive at Fenway will travel nearly 400 ft at Coors, certainly enough to make the difference between a warning path flyout and a home run. By the way, the Colorado Rockies claim on their web site that the difference is 9%. I disupt that claim.

What About That Humidor?

Starting in 2002, the Colorado Rockies started storing their baseballs in a controlled environment at 70 degrees F and 50% relative humidity. There was an immediate and significant reduction in the number of home runs. For the definitive research on this subject, see Home Runs and Humidors: Is There a Connection? an article I wrote for Baseball Prospectus. Be sure to read all the comments. For Coors, the predicted reduction in home runs is 27.5 +/- 4.3 %, in nearly perfect agreement with the actual 25% reduction. For Chase the predicted reduction is 37.0 +/- 6.5 %.

I did an interview with Andrea Seabrook regarding both the effect of altitude and the storage of baseballs in a humidor. See the NPR Broadcast from Saturday October 27, 2007 on All Things Considered entitled "Baseball Humidor Aids Fair Play in Denver."

How Much Does a Fly Ball "Carry"?

Using hitf/x and hittracker data, I have recently developed a numerical measure of the "carry" of a fly ball for each ball park. In a vacuum, all balls hit with the same initial velocity and launch angle will travel the same distance. In reality, the ball will travel more or less than that, depending on the influence of the aerodynamic effects of drag and the Magnus force, including any influence of wind. A measure of the "carry" is the ratio of the actual distance to the distance it would have traveled in a vacuum. In this plot , I have analyzed 819 home runs from the first 6 weeks of the 2009 season and determined the carry for each park. Actually shown is a "normalized carry", which is the actual carry divided by the average over all ball parks (so that the mean normalized carry is necessarily 1). The striking thing about the plot is that Coors, highlighted in green, is head and shoulders above all the other ball parks, with a carry about 7.5% larger than average. Roughly speaking this corresponds to an extra 30 ft on a home run relative to the average. On the opposite end of the spectrum, Progressive Field in Cleveland, highlighted in cyan, has a carry about 4% lower than the average. Finally, the new Yankee Stadium, highlighted in purple, has a carry about 2% below average. If home runs there were significantly aided by the wind, one might expect an above average carry. The fact that the carry is below average suggests that there is no evidence in the data analyzed for any significant effect of wind at the new Yankee Stadium. For a more detailed account of this analysis, see this link.

References