Home Runs and Ballparks

Baseball is unusual in that there is not a standard playing field. Outfield distances vary from 200 feet (Little League) to infinity (many high school fields). Major League outfield

distances vary from 302 feet to 435 feet and outfield fence heights vary from 4 feet to 37 feet. Given these variations, it is of interest to estimate the effect of Major League

ballpark configurations on the ease of hitting home runs.

Such an estimate is obtained by calculating

  the minimum energy required to hit a mid-July home run in the different ballparks

under a set of reasonably typical conditions for each ballpark. More specifically, the minimum energy for each ballpark is determined for home runs hit down the foul lines, the

power alleys, and to dead center. These five minimum energies are then averaged to provide a measure for each ballpark. A horizontal tail wind is arbitrarily directed toward

center field, as illustrated in Figure 1.

Distances to the outfield fences and the fence heights at the 30 ballparks are listed in Table 1

.  On the average, the American

League distances are significantly shorter to the left field, right-center field, and right field. However, only the average left field fence is significantly higher, primarily due to

the Red Sox green monster.

In determining the minimum energy required to hit a home run, forces on the baseball must be computed as the ball travels to the outfield fence. These forces, indicated in

Figure 2, are (1) the horizontal wind pushing the baseball toward the fence, (2) gravity which pulls the baseball toward the ground, (3) air drag which is the braking action of

the atmosphere, and (4) the Magnus force which results from the spin. The wind and gravity forces are constant during the flight of the baseball. Furthermore, the gravity

force is the same for all ballparks.

The air drag force is proportional to the air density, the velocity squared, and the drag coefficient. The drag coefficient is a constant 0.50 below 40 mph, decreasing to 0.29 at

120 mph. The high altitude Rockies have the lowest value air density at 0.79 atmospheres. The air density of the other ballparks varies from the sea level value of 1.00

atmospheres (several low elevation teams) to 0.90 atmospheres (Diamondbacks). The air drag force is always in direct opposition to the velocity.

Batted balls usually have appreciable backspin, which is fixed here at 17.60 times the initial velocity, e.g., a 100 mph baseball would have an initial spin rate of 1760 rpm. Due

to wind resistance, this rate is slowed throughout the flight. As an estimate of this decrease, the spin is linearly ramped to zero in five seconds, although the ball may continue

in flight.

The spin causes a pressure differential on opposite sides of the baseball that deflects the baseball in the direction perpendicular to both the velocity and the spin axis. This

Magnus force, which acts to curve thrown balls, depends on the spin rate, the velocity, and the drag coefficient. For a pure backspin with the spin axis parallel to the ground,

the vertical component of this force increases the height of the trajectory throughout its flight by increasing the vertical velocity, as shown in Figure 3. However, until the ball

reaches its maximum height, the horizontal component of the Magnus force decreases the horizontal velocity. After the ball reaches its maximum height the horizontal

component of the Magnus force increases the horizontal velocity for the remainder of the flight, albeit with a reduced effect since the velocity and spin rate are less on the

downward flight than on the upward flight. The net effect is to decrease the minimum energy by about 2%.

At the beginning of the baseball’s flight, the gravity and air drag force are roughly the same, with the Magnus force being about 75% less. The bat provides the main force,

but it only lasts for around 0.005 seconds. It propels the baseball at the initial angle relative to the ground. The calculations do not include the details of the force provided by

the bat or the compression of the baseball but effectively includes their effects by assuming an initial velocity of the baseball at a height of 39 inches above home plate. The

calculations vary the initial velocity and initial angle until the energy is minimized.

A typical optimized trajectory is shown in Figure 4 for a home run hit along the right field foul line of the Pirates ballpark. The initial velocity is 95.4 mph at a 40.0-degree

angle. At 320 feet the ball just clears the 21-foot fence (21 feet to honor the great Pirates right fielder, Roberto Clemente, who wore the number 21). This figure also shows

how the velocity decreases over the first 220 feet, due primarily to air drag, and then increases at longer distances, due to gravity, until it is 65.2 mph at the outfield fence.

The home run results which gave the lowest average hit energy show that the optimum hit angle varies from 36.6 degrees (Diamondbacks) to 41.7 degrees (Red Sox). The

optimum initial ball velocity, to just clear the outfield fence, varies from 98 mph (Red Sox) to 110 mph (Brewers). The shortest flight time is for the Angels (4.23 seconds) and

the longest flight time is for the Nationals (4.56 seconds).

The calculations produce the energy expended from the time the ball is hit until it clears the fence. However, most of the energy expended by the batter is lost in friction. This

effect is roughly estimated by multiplying the calculated energy by 4.7. The final result is then an estimate of the minimum energy that a batter has to expend to hit a home

run in one of the directions in Figure 1. The final energy values are in units of ft-lbs. To put the values in more understandable units, the energies are expressed in terms of

how many pounds of weight a person would have to lift 30 inches (a typical bench press length) to expend the determined energy in ft-lbs.

The minimum energy results, averaged over the five directions, for all thirty ballparks are given in Table 2. Included are the city-dependent parameters.  The city elevations

are from the pmiusa web site. City temperatures are mean July values given at the National Climatic Data Center web site. Elevations and temperatures are needed to

calculate the air density

.  Each wind speed is the ground level July values given in the Wind Energy Resource Atlas. Wind speeds in

parentheses are for the city, but the calculations assume the wind speed for these ballparks is 0 mph since they are enclosed.

The energy results show a 19% difference between the lowest and highest values (only a 10% difference if enclosed ball parks are excluded). The Red Sox (short outfields)

and Rockies (low air density) have the easiest ballparks. The hardest ballparks in which to hit home runs are enclosed ballparks where there is no wind
Temperature has a small effect; when the Diamondbacks temperature is reduced by 20 degrees, the required initial velocity is only increased by 1.0% and the energy

increased by 2.2% due to the greater density of cool air. However, wind has a significant effect; when the Giants 13 mph wind is reduced to 0 mph, the required initial velocity

is increased by 7.7% and the energy by 16.0%. This effect explains why it is easier to hit home runs with even a small tail wind in an open-air ballpark than in an enclosed

ballpark (even ballparks with retractable roofs have such high supporting walls that there will be little wind on the field).

The easiest home run (162 lbs) is along the Red Sox right field foul line (shortest distance) and the hardest home run (329 lbs) is to the Astros center field (longest distance).

The longest measured home run (565 ft) was hit by Mickey Mantle in 1953 at the Griffith Stadium in Washington D.C. With the recorded steady wind of 20 mph, this home run

would require a lift weight of 408 lbs – 19% more than needed for an Astros center field home run and the initial baseball velocity would have been 145 mph! These high

values cast some doubt on the authenticity of this record.

The results indicate that most of the ballparks have about the same degree of difficulty. They also indicate that Barry Bonds hit primarily in an easy ballpark, Mark McGwire in

an average ball park, and Sammy Sosa in a slightly harder ballpark (although the Cubs ballpark is famous for its variable winds).

This analysis gives estimates of relative difficulties in hitting home runs during a typical July day. It is based entirely on the ballpark dimensions and locations, plus nominal

temperatures and wind velocities. The analysis could be improved if actual wind values and directions were available for each ballpark. However, relevance to actual home

run production will always be limited by managers who routinely alter their lineups and strategies to accommodate ballpark conditions.

Additional subjects for discussion are provided by the 2007 Major League hit distribution given in the 2008 ESPN Baseball Encyclopedia and reproduced in Table 3. In spite of

the designated hitter in the American League, the percentage of home runs is essentially the same in both leagues – although the percentage of singles is slightly higher.

However, the most striking feature is that Major League outfield fences make it five times easier to hit a home run than a triple. The big mystery is therefore why do triples

count less than home runs? Furthermore, why do rare inside-the-park home runs only count as much as the much easier outside-the-park home runs?

¹The baseball trajectory equations are given by R. K. Adair in his book: The Physics of Baseball.

²http://www.ballparks.com/baseball

³Humidity differences are not included in the calculations. Humid air is less dense than dry air but humid conditions also increase the baseball weight and
inelasticity. The net effect is that humidity decreases the flight distance slightly.