Can a baseball be hit farther at high altitude?

bibliophage replies:

You might not like this, slugger, but I gotta call ’em like I sees ’em.

[A]tmospheric pressure, which covaries with altitude, pushes in on the ball from every direction, effectively canceling itself out.

Strike one. Try this experiment. Stand on the roof of your car while your friend drives along the highway at 110 m.p.h. Now tell me that the air pushes you in every direction equally. Oops. Didn’t see that overpass coming, did you? Try this instead. Stick your hand out the window of a moving car. Whoops. Telephone pole. Maybe you should leave the experiments to someone else. The bit you learned in high school physics about air pressure pushing equally in all directions (assuming buoyancy is negligible) applies only when the object is not moving through the air (or, equivalently, when the air is not moving past the object). A baseball typically leaves the bat traveling about 110 MPH. Air has mass, so the ball has to push the air out of the way to continue on its trajectory, and the air pushes back. The amount of work the ball has to do is proportional to the mass of air it has to move, and that’s proportional to the density of the air. (Air pressure is related but not identical to the density). Your friend has to stomp on the accelerator a little harder to maintain 110 MPH with you on the roof, but the ball doesn’t have that kind of power source. So a batted ball in air slows down a lot, which reduces its range. A ball that would go 400 feet in air at sea level would go about 750 feet in a vacuum, just because it doesn’t have to push the air out of the way. The air density in Denver is about 18% less than at sea level (assuming the temperature and humidity are the same), which works out to a 10% increase in range, or an extra 40 feet.

[A]ir pressure co-varies with weather patterns, too, but you never hear anyone talking about how it’s easier to hit homeruns on windy, low pressure days.

Strike two. I don’t know who you’ve been listening to, but where I come from, baseball commentators frequently refer to the effect of weather on the likelihood of hitting the long ball. If you’re playing in the open, a 10 MPH tailwind can add 30 feet to the range (and a similar headwind can deduct 30 feet). Obviously, the stands at major league ballparks act as a windbreak, but they don’t eliminate the wind. Temperature is another factor, because hot air is less dense than cold air. Due to air density alone, a ball will go about 20 feet farther on a hot summer day of 90 degrees F than on a cold spring evening of 40 F. The temperature of the ball augments the effect (perhaps as much as doubling it), because warm balls are livelier (bouncier) than cold balls and so will go farther. Humidity is a complicating factor. Humid air is less dense than dry air, but on the other hand, high humidity makes the balls deader (less bouncy).

[A]t Coors Field, you’re just enough further from the center of the Earth to make home runs comparatively easy to hit.

Strike three, you’re out. On the surface of the earth, gravity doesn’t vary enough to be a major advantage to batters at any ballpark. Even if it did, the Colorado Rockies wouldn’t be the beneficiaries. Remember that the earth is not a perfect sphere–sea level at the equator is 13 miles farther from the center of the earth than sea level at the north pole. When you also take into account centrifugal force from the rotation of the earth (which is greatest at the equator) this further reduces the gravity of the lower latitudes. Neglecting gravitational anomalies, I calculate gravity at Miami to be weaker than in Denver, despite Denver’s higher elevation, because of Miami’s lower latitude. But the effect is small, less than one part per thousand. The gravity at Seattle is a little stronger than at Denver, by a little more than one part in a thousand. Philadelphia, which is at about the same latitude as Denver, has stronger gravity, owing entirely to the difference in elevation, but only by about a half part per thousand. All these differences in gravity are negligible when compared to the effect of air density and wind. A ball might go one foot farther in Miami than in Seattle because of gravity, all else being equal. That’s not much to speak of compared to the 40 feet difference due to Denver’s thin air.

By the way, the oblateness of the earth leads to the interesting fact that although Mt. Everest is the highest mountain in the world, as measured relative to sea level, it is not the surface point farthest from the center of the earth. Chimborazo, a volcano in Ecuador (the same one as in the poem “Romance” by Turner) is about 8,000 feet shorter (relative to sea level) but because it’s much closer to the equator, its peak is farther from the center of the earth. The gravity there is almost 4 parts per thousand weaker than in Denver, so I would weigh almost a pound less there. Chimborazo, Cotopaxi, they had stolen my weight away!

Undoubtedly Coors Field is a hitter’s park, but there’s more to it than just the fact that a batted ball goes farther in the thinner air. Pitchers are at a distinct disadvantage because they lack the same degree of control there. All pitches that depend on the aerodynamic properties of the spinning baseball (and that’s most of them), are harder to throw well. The curve ball, for example, will curve only about two-thirds as much in Denver’s thinner air. Balls batted toward right or left field, because they pick up sidespin when hit, tend to curve toward the outside by several tens of feet for the same reason that curve balls curve (the Magnus effect). Again, at higher elevation, the effect is weaker, so many balls that would curve foul at sea level stay fair in Denver. Another factor to consider is that a ball of a given distance doesn’t stay in the air as long in thinner air, so a fielder doesn’t have as much time to get to it.

But none of this would seem to give the Rockies an unfair advantage in terms of winning games, since their opponents have the same advantages and disadvantages when playing at Coors Field. The only real concern is that the thin air will make Rockies batters look better than they really are, and pitchers look worse than they really are. When it comes time to consider Rockies players for inclusion in the Hall of Fame, no doubt this will be taken into account. Luckily, there exist a pair of statistics, called “park factors” (one for batters and one for pitchers) that account for whether a park is a pitchers’ park or a hitters’ park. (I won’t go into details, because it made my head hurt when I tried to understand it, but these statistics are calculated based on three years of data, and they do vary somewhat over time even at the same park). The higher the number of either statistic, the better the park is for hitters. Coors Field and Mile High Stadium (where the Rockies played before Coors Field was finished) have by far the highest park factors in majors, the factor for batters having been as high as 131 and for pitchers as high as 129 (100 is average). Over the last 100 years, no other park in the majors has had a park factor higher than 115 (both for batters and for pitchers, at Wrigley Field in 1969-71).

Even with their obvious advantages, the Rockies haven’t racked up as many home runs as you might expect. They have hit as many as 239 homers in a year (in 1997). That’s respectable, but only good enough for second place in the history of the National League (the Astros hit 249 in 2000) and tenth in the majors (the 1997 Mariners are first with 264, but in the American League where the designated hitter is allowed). The Rockies have scored as many as 968 runs in a season (in 2000), the most for any National League team since 1930, but nowhere near the record of 1220 runs (in only 133 games) set in 1894 by the Boston Beaneaters. (Now there’s a team you don’t want to be stuck on the bus with.) The Beaneaters later became the Boston Braves, and eventually the Atlanta Braves. I can’t imagine why they changed the name.

Starting this year, the Rockies (who supply all the balls for games at Coors Field) are storing their balls in a humidor. This should make the balls deader (less bouncy). If it works (and it’s too soon to say for sure), it should make the pitching staff happy by reducing their mile-high ERA to a more reasonable figure. For example, the Rockies managed to win about four-ninths of their games in 1999, even though their pitching staff had a combined Earned Run Average of 6.02. Since 1940, only the luckless ’96 Tigers have had a higher ERA, and they won fewer than a third of their games. But again, since the Rockies and their opponents will use the same humidified balls, no team should have an advantage in terms of winning games.

So there you have it. Now there’s only one thing in baseball I can’t figure out. How in hell can a man with four balls walk? Sounds awfully painful to me.

Further Reading:

The Physics of Baseball by Robert K. Adair

bibliophage

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