'Baseball's a simple game. You hit the ball. You throw the ball. You catch the ball,' said a well-respected baseball manager by the name of Casey Stengel. Mr. Stengel was a baseball man, not a mathematician nor a physicist.

Physics and mathematics can be applied to the game of baseball on every pitch, and on every swing of the bat. To understand the physics of the game, it is first necessary to look at the center of the game, the ball. Section 1. 09 of the Official Baseball Rules states that the ball must weigh between 5 ounces and 5 1/4 ounces, and that the circumference of the ball must be between 9 inches and 9 1/4 - inches (web). The velocity of the ball plays a large part in its motion. When the ball is traveling at a speed of about 50 miles per hour or less (small velocity), it is said that the air runs "smooth" over the ball, which does not create much movement.

For velocities of about 200-mph or more, the air surrounding the ball, and the air trailing the ball, is said to be quite "turbulent" (Adair 6). However, for the most part, the game is played with velocities between these two areas, which creates a gray area where characteristics of both can be observed. When a ball is hurled towards home plate by a pitcher, it can be forced to move in different directions if there is an altered surface on the ball traveling at a small velocity. This can be achieved by illegally placing a foreign material, such as spit or Vaseline, onto the ball. Movement can also be achieved when a ball is changed through use during the game  to prevent such movement, balls are changed constantly throughout the game.

The air resistance is, surprisingly, smaller for turbulent air than for smooth air. Despite popular belief the biggest opponent that a hitter faces is not the pitcher it is air resistance. If a ball were hit with a velocity of 110-mph at an angle of 35 , it is expected to travel about 700 feet, if it were hit in a vacuum. However, baseball is not played in a vacuum, and a ball with those characteristics would only travel about 400 feet.

The force that is placed on the ball depends on the velocity of the ball and the drag coefficient, which varies slowly with the velocity (Adair 6). In the graph below, the drag coefficient for a baseball hit at 110-mph is about 0. 2. Because the mass of the ball is constant, and the air density does not vary much for the conditions where baseball is played, the force on the ball is proportional to the velocity squared times the drag coefficient. The rotation of the ball has a small effect on the forces against the ball. If a ball is traveling with a high rotation rate, the drag will increase about one percent of the drag, which does not make a significant difference in the velocity of the ball as it crosses home plate.

(web) In most major league ballparks, the density of the air is relatively equal. However, in the hitter friendly confines of Coors Field in Denver, Colorado, home of the Colorado Rockies, the air density does play a major role in the distance the ball travels. A ball hit in Shea Stadium in New York by Mets power-hitter Mike Piazza that lands 400 feet from home plate could travel up to 40 feet further in Coors Field. This has caused a lot more home runs to be hit in Colorado, which excites the fans and hitters, but is hated by the pitchers (Adair 18). Several tests have been performed in order to help further understand the drag on the ball. When a ball is placed in a wind tunnel with an upward wind velocity of 95-mph, the ball appears nearly motionless.

This has lead researchers to conclude that for a ball traveling at 95-mph, the drag on the ball is equal to its weight. This is why throwing the ball at a higher initial velocity is beneficial to the pitcher. Not only does the ball reach the plate in a faster time, but also the drag on the ball is less so the ball will have a faster velocity as it approaches the batter (Adair 9). The graph above shows the drag coefficient for a rough ball, a smooth ball, and a baseball with different velocities.

Baseballs are usually pitched at a velocity of about 90-mph, which translates into a 0. 3 drag coefficient. If a ball were thrown straight the batter would have no trouble hitting it. Therefore, the pitcher must change different place additional forces in order to make the batter miss. Different pitches have different properties because of the forces that are placed on them. The spin and the velocity that is placed on the ball by the pitcher control most pitches.

In order to throw a certain pitch, a pitcher must place a certain spin on the ball. Also, the arm angle from which the ball is thrown plays a part as to where to the ball will cross the plate. Each pitch is designed to fool the hitter into thinking that the pitch will end up in one place, when it actually "dives" to another place. Legendary Boston Red Sox hitter Ted Williams often describes how he was able to see the spin on the ball, enabling him to determine the type of pitch and therefore, the location it was bound to move in. The following diagram shows the different spins for different pitches from a right-handed pitcher, as seen from the hitter's point of view.

The Magnus Force is also moving in the direction of the arrows. (web) Take four pitches -- the fast ball, the curve, the slider and the screwball. Now throw these at different speeds, and you have twelve pitches. Next, throw each of these twelve pitches with a long armed or short armed motion, and you have twenty-four pitches. -Ed Lop at, Yankees pitcher, 1948-1955 (Rubin 43) The most basic and probably the most dramatic pitch is baseball is the fastball. Its name suggests that the pitcher attempts to throw the ball as hard as he can.

However, upon its arrival at the plate, the ball appears to "hop" about four to five inches. This may not be that significant of a movement, but if the batter were not to compensate for this change, he would completely miss the pitch because he must start his swing before this movement ever occurs. In fact, half of the fastball's movement will occur in the last 15 feet of its 60 foot-6 inch flight (the distance between the pitcher and home plate). This hop occurs because the tremendous backspin that accompanies the pitch. The ball curves because of the unbalanced force know as the Magnus Force. Professor Robert Adair of Yale University, and author of The Physics of Baseball, describes the Magnus Force in the equation: Magnus = KwVCvWhere: K = Magnus Coefficient (2 10-6); w = rotation of the ball (measure in rotations per minute); V = velocity of the ball (measured in miles per hour); Cv = drag coefficient.

(web) The backspin creates a Magnus Force that drives the ball in an upward direction. But because the pitcher can grip the ball in different orientations, different Magnus Forces can occur because a different number of stitches can pass the axis in any given rotation. One type of fastball is the 2-seam fastball, or the with-the-seams fastball. In order to throw this pitch, the pitcher must grip that ball "with the seams" as the name suggests. It will show two seams per rotation, and therefore will have a greater drag force. This creates a large Magnus Force, and a larger "hop." Another type of fastball is the four-seam fastball, or cross-seam fastball.

This pitch presents four seams per revolution and therefore a smaller a drag. In fact, the four-seam fastball can travel up to 2-mph faster than the two-seam, which means that the ball will reach the batter six inches faster. However, the downside to the four-seam fastball is that it often produces a small Magnus Force which does not allow it to "hop" as much as the two-seam fastball. The following diagram shows how a pitcher must grip the ball in order to throw the two fastballs described. The green lines represent fingers of the pitcher. (web) A third type of fastball that can be thrown is the split finger fastball.

To throw the splitter, the pitcher grips the ball like a four-seam fastball, but places his fingers on the smooth part of the ball, rather than the stitches. This allows the ball to slip off the fingers easier, thus creating less rotation. This causes the Magnus Force to decrease even more than the four-seam fastball, causing this pitch to sink. Although the spitball is illegal, the physics behind it are quite simple to explain.

The pitch is thrown with a split finger grip, but an extra lubricant is placed on the pitcher's fingers in order to allow the ball to be released with even less rotation. This causes the ball to have very little rotation and a high velocity, like the fastball. This creates a devastating pitch that is unpredictable, and that is why it is illegal to throw (web). The most complicated and most studied pitch of all is the curve ball. Physicists have studied for decades in an attempt to determine what makes the curve ball curve.

The principles of the curve ball can be traced as far back as 1671, when Isaac Newton observed a tennis ball. Newton observed that when a tennis ball was hit, it had a tendency to curve. This has been the basis for the study of the curve ball and its properties (Watt 39). At first it was believed that the ball actually did not curve, and that it was an optical illusion due to the spin placed on the ball by the pitcher. However, the development of fast photography has demonstrated that a curve ball does indeed curve. High-speed photography allowed pictures to be taken of the ball throughout its flight toward home plate.

The ball curves because of the unbalanced force know as the Magnus Force (web). When the pitcher places extra spin on the ball to create the effects of this pitch, the rotation and the stitches on the ball will cause one side of the ball to have less pressure than the other side. This causes the ball to move faster on one side than the other does, which leads to the curve of the ball. In order to throw the curve ball, a pitcher must place spin on the ball that is not parallel with the ground, but perpendicular, in order to create this effect. In Coors Field, pitchers are able to notice that their curve ball will not "bite" as much as it would if they were in any other park in the league. This also has to do with the air density in the city of Denver, which is 5, 280 feet above sea level (Adair 18).

Another complex pitch that is studied is the knuckle ball. Any catcher will tell you that a knuckle ball may be the hardest pitch to catch because of its unpredictable movement. The pitcher attempts to place as little rotation on the ball as possible, so that the ball is, in theory, floating towards the plate. This pitch is thrown with a very low initial velocity and therefore has large drag force acting on it.

The large drag force causes the ball to move unpredictably, and often has late-breaking motion that creates a problem for both the catcher and batter. Any small change in orientation in the way the ball is gripped or thrown can change the trajectory of the pitch and cause it to travel in an undesirable path. The reason that the knuckle ball is able to change directions during its flight toward the plate is that the stitch orientation changes because of the lack of rotation. A regularly thrown fastball has a constant stitch orientation because the ball is thrown with a high velocity that keeps the ball rotating in a constant path. But the knuckle ball is thrown with such a small velocity, and a lack of force to put the ball in rotation, that the ball is able to "show" more than one stitch orientation on its way to the plate. This causes the Magnus Force to constantly change directions, and therefore, "push" the ball in different directions.

If the ball does not change directions, then it will be easily hit because of the ball's lack of rotation and velocity. Despite some inconsistencies, some pitchers are able to master the technique of the knuckle ball, and become effective pitchers with an uncontrollable, and usually devastating pitch (Adair 30). Another aspect of the game that can be analyzed by physics is hitting. Hitting is an art that is very difficult to master.

Major League Baseball rules state that the bat may be no more than 2 3/4 inches wide at the thickest part, and that the bat may not exceed 42 inches in length (web). There are different opinions on the topic of whether a heavy or light bat is better for hitting. A light bat will allow the hitter to swing faster, which gives him extra time to watch the pitch as it comes toward him. However, a light bat also reduces some of the power that would accompany a heavier bat. A heavy bat can give the batter more power with his swing according to the equation: momentum = mass velocity However, if the batter cannot achieve sufficient velocity in his swing, then the use of a heavy bat is irrelevant because the momentum that would be gained by using a heavy bat would be lost (web). Taller batters must use longer bats, which means that their bats are heavier than shorter players are.

Since research has been conducted on the game, scientists have attempted to find a place on the bat that excess weight could be removed without altering the hitting surface. Through experimental research, it was discovered that on the top of the bat, above the barrel, was where the most weight could be removed without changing the active part of the bat. Players were able to alter their bats by sanding down a circle on top of their bats, which cut down on some excess weight. This allowed all players to make their bats lighter without cutting down on the actual hitting surface (web /physics pages/ Phys Of 1998 A/Baseball-Sakamoto / page 1.

htm). Up until recently, baseball has been a predominately a pitcher's game. Hitters have always looked to gain an extra, unfair, advantage against the pitchers. Aside from sanding out the top of the bat that helped the batter decrease bat weight even with increasing size, hitters have placed illegal substances inside their bat. Corking, as it is known, occurs when cork replaces some of the wood in the middle of the bat, causing the ball to "spring" off the bat and increases the distance the ball would travel. However, a corked bat is weaker than a regular bat, which would cause the bat to break much easier (web).

If the bat splits open, the cork will be exposed and the player will be suspended for using illegal equipment. Also, the corked bat has traditionally been used by smaller players, not known for their hitting abilities. The last player found using a corked bat had a. 211 batting average, not a very good average at all. Using a corked bat can improve a batter's power swing, but in more often than not, it is either detrimental or has no effect (web).

In order to determine how far a ball will travel, it is important to understand the coefficient of restitution. This is a physics term that describes what happens to the bat and the ball, before and after contact is made. It describes how much energy the ball retains after contact with the bat. If this coefficient is high, then the energy from the bat and ball is retained, and the ball will have more energy after impact. If the coefficient is low, then some of the energy has been absorbed by the collision, and the ball will not travel as far (web).

In order to obtain the largest possible coefficient, the batter must make contact with the ball on the "sweet spot" of the bat. By effectively hitting this spot, more energy is conserved than on any of the part of the bat, and causes the ball to be hit further and faster. A hitter can feel when he has hit the sweet spot because it feels as if he did not even hit the ball. That is why players such as Mark McGwire and Sammy Sosa are able to know they have hit a home run even before the ball has cleared the fence (web). Another question that has been studied is how these tremendous athletes are able to generate such power out of their bat. Hitters are taught to "step into the pitch;" that is, transfer their weight from their back foot to their front foot as they swing.

During this process, the hands and the bat can reach a velocity of about 70-mph. Because the bat is moving at such a high speed, even a difference of 0. 01 seconds can determine if the ball will be a home run or a pop up, or a fair or foul ball (Adair 50). The placement of the batter's hands also plays a part in the power that can be generated from the swing.

To optimize the power in the bat, the batter can hold the bat at the very end. However, in doing this, more time is required to get the bat over the plate to hit the ball at the right time. So a player must commit himself earlier to the pitch, which could mean that he swings too early if the ball is not thrown at the pitcher's normal speed, known as an off-speed pitch. The equation below shows that a higher velocity can be achieved by having a larger swinging radius: Velocity of bat = velocity of swing radius of swing Holding the bat closer to the barrel, or "choking-up," does not generate as much power, as seen in the equation above, but it allows for a shorter swing so the batter has longer to decide if the pitch will be a strike or a ball. Smaller players usually "choke-up" to gain greater control over the bat.

Players such as Pee Wee Reese and Chuck Knoblauch usually hit for average rather than power (web). The way the batter swings also can determine the outcome of the swing. The ball comes toward the batter at an angle of about 10 . If the batter were to hit the ball with a swing that had the same 10 angle, the ball would travel in a line drive. Players with these types of swings, such as Rod Carew and Wade Boggs, often hit for very high averages with very few home runs. They are able to hit the ball with great precision and can usually time the pitch so that the can hit the ball right in front of the plate.

This type of hitter is usually very good with directing the flight of the ball. They are able to time their swing so they can hit the ball to either side of the field by changing their timing (Adair 70). If the ball were hit nine inches early, the ball would be "pulled." If the batter were to "wait" on the pitch, and swing late, then the ball would be hit to the opposite field (Adair 70). Contrary to Carew and Boggs, Hall of Famers Reggie Jackson and Harmon Kille brew hit the ball with an uppercut swing. This type of swing usually meets the ball coming in at 10 with a bat angle of about 25 . While Carew and Boggs hit for line drives, the uppercut swing of Jackson and other power hitters was meant to drive the ball into the air with tremendous power.

If these two types of swings were to hit the same ball with the same power, Carew's ball would land between the outfielders for a double, while Jackson's ball would end up in the bleachers for a home run at Yankee Stadium (Adair 70). The national pastime, as baseball is known, has been studied and picked apart by researchers and scientists. Both pitching and hitting philosophies can be explained through the complicated terms of physics. Everything from a curve ball to a towering home run has some driving force behind it that can only be explained through the in-depth study of the physics of the complicated, yet so simple, game of baseball. Mr. Stengel had the right idea.

Works Cited Adair, Robert K. The Physics of Baseball. New York: Harper & Row Publishers, 1990. Conley, Ken; Chang Choi, and Joe Giuliani. "Baseball: The Game and Beyond." Online. Internet.

Available WWW: web Richard. "The Physics of Baseball Pitching." Online. Internet. Available WWW: web League Baseball. "Rules of Baseball." Online. Internet.

Available WWW: web Monte leone, John and Mark Gola. The Louisville Slugger Ultimate Book of Hitting. New York: Henry Holt & Company, Incorporated, 1997. Rubin, Louis D.

"The Quotable Baseball Fanatic." New York: The Lion Press, 2000. Sakamoto, Garrett. "The Physics of Hitting a Baseball." Online. Internet.

Available WWW: web Tom. "Physics of Sports Lecture 18." Online. Internet. Available WWW: web Robert G. and A. Terry Ba hill.

Keep Your Eye on the Ball: The Science and Folklore of Baseball. New York: W. H. Freeman and Company, 1990.