THE PHYSICS OF BASEBALL

Raw Material for Lectures Given by Porter Johnson at Various Chicago Area High Schools

Baseball is our "national game", which was reportedly invented by Abner Doubleday in New York in the 1840's. However, there is possible truth in the suggestion that Doubleday communicated with Russian immigrants concerning the game LATPA, as did the Englishmen who invented cricket---since otherwise one is at a loss to understand the formal relation of baseball to other "stick ball"sports. By the end of our Civil War the soldiers on both sides were becoming increasingly pre-occupied with baseball. These activities caught the eye of a young Civil War correspondent named Walt Whitman, who became our greatest poet, and who had the following to say about baseball:

I see great things in baseball. It's our game, the American game. It will repair our losses and be a blessing to us.

Professional teams were soon organized, and the National League was officially begun in 1876. Networks of teams were formed in cities throughout the land, and the World Series between winning teams in the American League and the National League began officially in 1903. Ty Cobb, The Georgia Peach honed his spikes to razor sharpness on the sidelines, and re-fought the Civil War on the base paths in the early part of this century. Babe Ruth, released from a "reform school" in Baltimore, became the leading pitcher in the American League, and in the "live ball" era became the greatest home run hitter of all time for the New York Yankees, who dominated the game for several decades with the likes of Crosetti, Dimaggio, Gehrig, Dickey, Lazzeri, Musiel, Ruffling---followed by the equally awesome Berra, Ford, Kubek, Larson, Lopat, MacDougal, Mantle, Maris, Martin, Power, Raschi, Rizzuto, Skowron, and Turley. In baseball the children of the immigrants, the poor, and the downtrodden could rise to the top of popular culture. Shoeless Joe Jackson from Tennessee, who could neither read nor write, was banned from baseball after the infamous scandal in which Chicago White Sox players were accused of "fixing" the 1919 World Series. The boy chasing after him with tears in his eyes, saying "say it ain't true, Joe" is one of the most haunting images in sports history.

In the 1940's the nation could no longer ignore the fact that players in the Negro Leagues were often better than those in the major leagues, and were kept out of the majors by pure prejudice. The sport was integrated in 1947 when Jackie Robinson joined the Brooklyn Dodgers. A pantheon of heroes followed him---Satchel Paige, Don Newcombe, Roy Campanella, Willie Mays, Hank Aaron, and Ernie Banks, to name but a few---and baseball was forever enriched by their dedication and excellence. In a similar spirit, a wave of Hispanic players came in the 1950's---Minnie Minoso, Roberto Clemente, and a host of others---perhaps their English was a little quaint at times, but they knew how to play baseball. In the 1960's the nation agonized over whether Sandy Koufax, would pitch in the World Series on Yom Kippur. One could develop a fairly complete history of our nation over the last 150 years by studying the changes in baseball.

There is a direct and obvious connection of Illinois Institute of Technology and baseball, in that, as far as I am aware, our location to the White Sox Park is closer than that of any other university to a major league ball park. Furthermore, a long list of stars have used the facilities at IIT for training and practice---Tom Hanks, Bo Jackson, Michael Jordan, and Madonna! We think of the Chicago White Sox as an "IIT Baseball Team", although we do have many confused North-Siders in our midst!

Baseball is our national sport, but it is not only that. We are a meltingpot, an agglomerate of cultures, and not merely just another nation; and, in the words of Abraham Lincoln, we represent

"the last, best hope for mankind".

Our national sport of baseball is correspondingly international in scope--just ask supporters of the powerful Toronto Bluejays! Indeed, one precursor of the 1962 Cuban Missile Crisis was the fact that U-2 overflights showed soccer fields under construction in Cuba---for Russian advisors, since Cubans prefer to play baseball. Indeed, the Cuban team in the 1992 Olympics was considered unbeatable, with a team of "major league caliber" players, as cheered on by its greatest fan, Fidel Castro. Is it too naive to hope that, someday, the glory of nations will be determined on the sports field, instead of on the battle field?

A few years ago an American all star team toured Japan and played a series of games---and lost to the Japanese team. In my opinion, this was not entirely due to the "gai-jin" strike zone or other forms of local favoritism, but to the skill and pin-point control of Japanese pitchers, to the team spirit and cooperation of the Japanese players, and to over-confidence on the part of our team. In a similar vein, the skilled, well-trained teams from Taiwan often win the Little League World Championships.

One learns about baseball by playing it, by watching it, and by endlessly analyzing the statistical information! In a way, one learns more from watching the foibles of minor league baseball, rather than the elegance and perfection of the major leagues---much in the same spirit as learning to appreciate the subtle differences in tastes of good wine by first studying the significant differences in taste in mediocre wines.

The film Bull Durham provides insight into the game with the parallel careers of the experienced catcher [on his way down!] and the fast young pitcher [on his way up!]. This film brings out some of the sociological and philosophical aspects of baseball, such as the fact that

"baseball requires nonlinear thinking".

One player advocates an amendment to the U.S. Constitution to outlaw the designated hitter and astroturf.

I will stick primarily to the Physics of Baseball.

The major league pitcher stands on the mound a distance somewhat less than 20 meters from the batter's box, and throws fastballs at a speed somewhat greater than 40 meters/second, so that the entire trip takes place in 0.4 to 0.5 seconds. The batter must decide in that brief instant whether, when, where, and how to swing. Because of the influence of gravity, the baseball drops significantly from a projected straight line path during its travel. The distance dropped is given by

d = 1/2 g t2 = 1.2 meters.

A batter must respond naturally and instinctively to the parabolic path for the baseball, since there is no time for him to "think" about where the baseball is going. Consequently, the batter must "sense" where the ball is headed, and decide whether to swing at it. It is remarkable to me as a scientist that the batter can hit such a thrown baseball. Certainly, a person off the street would have trouble even seeing a major league fastball, although a significant "hum" could be heard as it whizzes by! To illustrate the speed of reflexes needed to hit a baseball, drop a dollar bill [about 15 cm in length] through the fingers of another person, whose hand is resting on a table. That person has about 0.18 second to catch the bill, and very few people can catch it. Hitting a major league fastball is far more difficult!

Does a projectile curve?

That question was initially posed by Sir Isaac Newton, and he suggested the definitive experiment of placing three vertical poles in a row, and observing whether the ball travels on an "in-out-in" trajectory. Newton experimented with tennis balls and explained their curved path. Lord Rayleigh analyzed the path of a spinning ball quantitatively in the 19th century, and the curveball was firmly entrenched in baseball by the end of that century. There is no question that baseballs curve significantly in their travel toward the plate.

Why do baseballs curve? How much do they curve?

The curved path of a spinning baseball can be understood quantitatively through the Bernoulli effect, which also causes airplanes to fly and canvas covers on moving trucks and trailers to inflate. [One must design race cars so that air circulates under the vehicle to keep the wheels on the ground, since the speeds are much greater than those at which airplanes usually take off.] Actually, a more accurate and complete interpretation of the motion of a thrown baseball requires understanding the Magnus effect, which will be described presently. Curvature of the trajectory is much more pronounced in tennis, soccer, football, ping-pong, and even softball than in baseball---not to mention the frisbee!

A right-handed pitcher typically makes the ball spin [counter-clockwise!] about an upward vertical axis, and the speed of the air over the surface of the ball on the left side vL is greater than the speed on the right side vR . According to the Bernoulli effect, air pressure on the right side PR is greater than on the left side PL , and the net force causes the ball to curve to the left---away from a right-handed batter. The magnitude of the pressure differential is

DP = PR - PL = 1/2 rair [vL2 - vR2 ]

and the net force to the left [for laminar flow!] is

F = 1/6 DP A ,

where A is the area of the baseball. This sideward force is proportional to the density of air, the speed of the ball, and its angular velocity, and its surface area. Putting in a translational speed of 30 meters/sec and a rotation rate of 40 revolutions/sec, we obtain a force of about 0.1 Newtons, or about ten percent of the weight of the ball. The ball will thus curve by about ten percent of the amount that it drops under the influence of gravity, 10 - 20 cm. This estimate is consistent with the largest values measured in telemetry experiments---and somewhat at variance with estimates from hitters! Even though it may seem modest in proportion, the curve is difficult to hit because the batter must estimate the position and time of arrival of the baseball with accuracies 1 cm and 0.01 seconds, respectively to hit it well. As with the falling induced by gravity, the uniform acceleration of a "curving" baseball has the greatest effect when the ball is closest to the batter. Also, the spinning of the baseball makes its translational speed more difficult to estimate by the batter in the fraction-of-a-second that is available. In addition, the dynamics of the collision between the ball and the bat is affected by spin, just as in ping-pong and tennis.

If the baseball spins about a horizontal axis, it drops more or less than otherwise depending upon the sense of the spin, and the pitch is called a "drop" or a "rising fast ball", respectively. In the traditional "roundhouse curve" thrown by a right-handed pitcher, the axis of spin is about half-way between horizontal and vertical [toward first base], and the pitch curves down and away from a right-handed batter. [Actually, the extra downward drift is more detrimental to hitting than the horizontal motion!] The the axis of rotation of the "slider" is vertical, and it curves away from a right handed batter. Because of its rotation about the vertical axis, it is very difficult for the batter to distinguish it in flight from a fast ball---it appears to be moving much faster than it actually is. The "screw ball" rotates about an axis with downward tilt, and for a right-handed pitcher it curves into a right-handed hitter.

In its flight the baseball carries along a thin Prandl Layer of air, and one must take proper account of coupling of the ball to that air layer; the Magnus Effect. As speeds below 20 - 25 meters/second the air layer is inert, and the description based upon the Bernoulli principle is correct except for multiplication by a fixed geometrical drag coefficient. Above this speed, however, the air layer plays a dynamic role in changing the flow around the ball from smooth [laminar] to turbulent. The ball interacts strongly with the boundary layer, and it receives a sudden impulse when air vortices leave the ball. The pitcher can make use of the various exotic physical manifestations of this onset of turbulent flow. The "hopping fastball", the "sharp-breaking curveball", the "fluttering" and "drift" of the "knuckleball", and even the "hum" of the fastball can be explained qualitatively in terms of the transition to turbulence. The motion of the knuckleball, which hardly rotates at all, is dependent on the orientation of the stitches on the baseball, being highly erratic when thrown by one of the masters [Jim Konstanty or Hoyt Wilhelm] on one of their better days. In this region of transitional turbulence the behavior of the baseball is especially sensitive to features on the surface of the ball, as pitchers have discovered empirically over the last 150 years. The physics of the thrown baseball is not well understood at a detailed quantitative level, since the surface of the ball is neither "smooth" nor "rough", and in fact the seams are asymmetrically situated on the ball.

Mention should also be made of the "foreign substance" ball, or spitball as it was called in a less formal era, which moves erratically toward the plate. The center of mass of the ball is no longer at its geometric center, so that there is a slight "wobble". However, the primary effect is that the "surface imperfection" enhances the coupling of the ball to the boundary layer of air, resulting in erratic motion, just as with an illegally "scuffed" ball. It is very difficult to control the "spitball". The pitch was officially banned from baseball at the beginning of the "live ball" era---perhaps because well-pitched games are not very interesting from the viewpoint of spectators! Rumors persist that it is still being thrown by certain unscrupulous [but highly successful!] pitchers, but it is difficult to collect the evidence.

Why are baseballs hard to hit?

The probability that the batter gets a base hit [or better] from a "strike" pitch is about 0.1. Indeed, a batter gets three chances to hit the ball, before he strikes out---as happens especially to home-run hitters. A baseball is difficult to hit because the batter cannot follow the motion of the ball as it crosses the plate. More precisely, the brain cannot process the information in a timely fashion, and of order 0.1 seconds are required to make adjustments in the swing. It is travelling at a speed of 40 meters/second at a distance less of about 1 meter from his eyes, and thus it is moving across the field of view at a angular speed of 50 radians per second. This rate is comparable of the "repetition frequency" of the picture on a television screen [30 Hz], which cannot be independently resolved. At best a batter sees a blurred and extended image of the ball from the pitcher well before it reaches him---even when he turns around to face the pitcher in the bunt position. By contrast, the catcher always faces the pitcher directly, and almost always catches the ball [probability 0.99] since it does not cross his field of view. However, there is a significant chance that the catcher will miss a "wild pitch", merely because it is not coming right at him, so that he sees a blurred image.

Where on the bat should you hit the baseball?

It is a very bad idea to hit the baseball on the handle of the bat---even if the ball does not mash your fingers, since your hands will "sting". It is also wrong to hit the ball at the very end of the bat, since the ball will not go far and your hands will sting. In fact your hands will sting unless you hit the ball at the "sweet spot". This sweet spot or center of percussion of the bat is not merely the center of mass of the bat. It can be located by suspending the bat from the point between where you put your hands, determining the period of small oscillations, and measuring down a distance L equal to the length of an equivalent simple pendulum:

T = 2 p ÖL/g .

For T = 1 sec, the distance is L = 25 cm, corresponding to the "center of oscillation" of the bat. Incidentally, the force of the ball on the bat is substantial. If the collision between the ball and the bat takes place over 0.001 seconds, and if the ball keeps its speed of 40 meters/second and simply changes direction from "in" to "`out", the average force on the ball is 8000 Newtons, or more than 5000 times the weight of the ball. The reaction force can easily break the bat if the ball is hit at the wrong place.

How far do baseballs go?

If the baseball is given a speed v at an angle theta to the horizontal, its horizontal and vertical positions at time t are

x = v t cos q

and

y = v t sin q - 1/2 g t2 ,

respectively. At the optimal angle of launch, q = 45°, the ball travels a horizontal distance

L = v2 / g

and stays in the air a time

t0 = Ö2 * v/g .

Putting in the "fastball speed" of 40 meters/sec, we obtain L = 160 meters and t0 = 5.6 sec, corresponding to a "tape-measure" home run even in the metric Canadian fields.

Actually, this estimate is somewhat altered by the presence of air resistance. In fact, the optimal launch angle is 35 degrees to 40 degrees to the horizontal, the ball travels less far, and it spends less time in the air. Air resistance is negligible for pitching, but not in hitting; nor for throwing balls from the outfield. The resistive drag of a baseball moving at 40 meters/second is a force comparable in magnitude of the weight of the ball, which acts in the direction opposite to the velocity vector. It produces about a 10 percent "slowdown" in the speed of a pitched ball, but a 50 to 100 percent reduction in the distance travelled by a well-hit ball. At high altitudes [Denver, or better yet, Mexico City] the resistive drag on balls is reduced and they travel 10 percent further than elsewhere. Curve balls are also less pronounced in these hitters paradises.

How high do baseballs go?

If the ball is launched upward with the speed of 40 meters/sec, it will attain a height of 80 meters and stay in the air for 8 seconds, under neglect of air resistance. It is said that the legendary Babe Ruth would often hit high "pop-ups" that were lost in the sun by players. The Babe was fast for a big man, and could run halfway around the bases while the ball was in the air. If the fielder missed, it might be a home run. One does not know how many of the Babe's 714 homers were of the "inside-the-park" variety. If the ball is popped up in the vicinity of home plate, it is often difficult to catch because of spin on the ball, which is caused by the fact that the collision of the ball with the bat is a "glancing" one, and which results in substantial horizontal motion in the ball.

How does one cheat at baseball?

It is possible that some very effective mechanisms are kept confidential as "trade secrets". Others are more obvious---such as moving the fences at the home field "in" when one has a slugger on the team, or "out" when one has strong pitching. My favorite, to be used only when one has a weak hitting team and very strong pitching staff, such as that of the Cleveland Indians of the 1950's [Feller, Wynn, Garcia, and Score], is to immerse the baseballs in liquid nitrogen, being careful to remove them a few hours before the game so that the outside of the ball will thaw. The collision between the ball and bat becomes more inelastic, in spite of the fact that the surface of the ball appears normal, because the entire baseball [and bat] participates in the collision. It is not known whether this form of cheating has actually occurred. It has frequently been denied, and it is a risky strategy.

Baseball Strategy

It is remarkable that minor changes in the rules and procedures of baseball lead to rather large divergences in optimal strategy for winning at baseball. In the current era the National League is the "pitcher's league", whereas the American League is the "hitter's league"---and this difference has developed in part because in the American League there is a designated hitter, whereas in the National League the pitcher bats for himself. The legendary New York Yankees had a finely tuned strategy in their years of glory, during which it was "macho" for pitchers to pitch the complete game. The Yankees, had a few "scatback" players [Lazzeri and Crosetti, followed by Martin and Rizzuto] who would swing prematurely and hit as many foul balls as possible [especially in the late innings]. The exhausted pitcher would then have to face the home run hitters. The greatest foul ball hitter of all time was Luke Appling of our Chicago White Sox, who hailed from the Ozarks. This "foul ball strategy" has become less effective in the modern era of the "specialist relief pitcher", which [by coincidence?] began when teams were allowed to enlarge their rosters a bit. [Also, it is hard to hit foul balls with the "specialty pitches" of certain relievers.] The net effect was to speed up the games, promoting customer satisfaction.

Baseball has survived wars, the Depression, night games, the perfection of the slider, the Pete Rose scandal, expansion to the North, South, and West, astroturf, the designated hitter, player strikes, umpire strikes, and Steve Dahl's anti-disco day at Sox Park. It has even survived the Brooklyn Dodgers moving to Los Angeles---"the most unkindest cut of all". We look forward to its continued enrichment of our culture.

Author, radio philosopher, and keen observer of the human condition Jean Shepherd has remarked that, if he were a Captain picking out men for a "hopeless" mission, he would go through the platoon and choose the Chicago White Sox supporters, since they are psychologically prepared. That is an accurate historical view of the Sox, but in recent years things have changed somewhat! We hope that our team has a great season, since our own fortunes are linked with their success.

References

[1] Robert K. Adair, The Physics of Baseball, 2nd edition [Harper 1994]. ISBN 0-06-095047-1

[2] David F. Griffing, The Dynamics of Sports. 3rd edition [Dalog, Oxford Ohio, 1984]. ISBN 0-9618361-1-2

[3] The Physics of Sports, edited by Angelo Armenti, Jr. [American Institute of Physics 1992]. ISBN 0-88-318-946-1

Back to Web Page