Physics and Rules Of Baseball Game

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According to a well-known baseball manager named Casey Stengel, “You hit the ball. You throw the ball. You catch the ball.” Despite being a baseball expert and not a mathematician or physicist, Stengel recognized the application of physics and mathematics in baseball, specifically in every pitch and swing. To comprehend the physics involved in the game, it is crucial to examine the ball, which is at the core of baseball. As per Section 1.09 of the Official Baseball Rules, the ball must weigh between 5 ounces and 5 ounces, with a circumference ranging from 9 inches to 9 – inches. The ball’s velocity significantly impacts its movement. When the ball travels at a speed of approximately 50 miles per hour or less (considered a small velocity), it experiences minimal disruption as the air flows smoothly over it.

The air surrounding a ball traveling at velocities of 200-mph or higher, as well as the air trailing behind it, is considered turbulent. However, the game is mostly played with velocities between these two extremes, resulting in a gray area where both turbulent and smooth characteristics are present. When a pitcher throws a ball towards home plate, it can be made to move in different directions if there is an altered surface on the ball traveling at a low velocity. This can be accomplished by illegally applying foreign substances, like spit or Vaseline, onto the ball. Additionally, movement can occur as a ball is used during the game. To prevent such movement, baseballs are constantly replaced throughout the game. Surprisingly, turbulent air offers less air resistance compared to smooth air.

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Contrary to popular belief, the pitcher is not the biggest opponent a hitter faces; it is actually air resistance. If a ball is hit at a velocity of 110 mph and an angle of 35 degrees, it would be expected to travel around 700 feet in a vacuum. However, since baseball is not played in a vacuum, a ball with those characteristics would only travel about 400 feet. The force exerted on the ball depends on its velocity and the drag coefficient, which changes slowly with velocity.

In the graph below,, the drag coefficient for a baseball hit at 110 mph is approximately 0.2.

Since the ball’s mass remains constant and the air density does not vary significantly under typical playing conditions, formula here. The ball’s rotation has a minor impact on the forces acting against it. When a ball has a high rotation rate, percentage value here, but it does not significantly affect the ball’s velocity as it crosses home plate.

Coors Field in Denver, Colorado, home of the Colorado Rockies, stands out among most major league ballparks due to its significant impact on ball travel distance. Unlike other parks with similar air density, a hit from Mets power-hitter Mike Piazza that reaches 400 feet away from home plate at Shea Stadium in New York could potentially go up to an additional 40 feet further at Coors Field. Consequently, this results in a higher number of home runs in Colorado, which thrills fans and batters but poses challenges for pitchers. Extensive tests have been carried out to examine the drag effect on the ball. In one particular experiment conducted using a wind tunnel featuring a 95-mph upward wind velocity, the ball seemed almost motionless. This led researchers to conclude that when a ball travels at 95 mph, its drag is equivalent to its weight. Therefore, it proves advantageous for pitchers to throw the ball with greater initial velocity.

As the ball approaches the batter, its speed increases and there is less resistance to its movement. The accompanying graph displays the drag coefficient for various baseballs at different speeds, including those with smooth surfaces and rough surfaces. Typically, pitchers throw baseballs around 90 mph, resulting in a drag coefficient of 0.3. If the ball were thrown directly, it would be easy for the batter to hit. Consequently, pitchers employ additional forces and modify the ball’s trajectory to deceive the batter. Each pitch possesses distinct attributes due to specific forces acting upon it.

The main factors that control most pitches are the spin and velocity applied by the pitcher. To throw a specific pitch, the pitcher must apply a certain spin and angle to the ball. Each pitch aims to deceive the hitter by moving in a different direction than expected. Ted Williams, an iconic Boston Red Sox hitter, could see the spin on the ball and predict its location. The diagram below shows various spins for pitches from a right-handed pitcher as seen by the hitter. The arrows indicate the direction of movement due to Magnus Force.

The fastball is the most basic pitch in baseball and also the most dramatic. It is thrown with maximum force, causing the ball to appear to hop about four to five inches when it reaches the plate. This slight movement can be crucial in the batter’s ability to make contact with the pitch, as they must start their swing before this movement occurs. Interestingly, half of the fastball’s movement occurs within the last 15 feet of its 60 foot-6 inch flight.

The bounce of the ball during pitching is caused by the strong backspin, which in turn generates a force known as the Magnus Force. Professor Robert Adair from Yale University explains this phenomenon in his book The Physics of Baseball. The Magnus Force propels the ball upward and its strength is affected by how the pitcher holds the ball. By gripping it with different orientations, the number of stitches passing the axis changes, resulting in varying Magnus Forces. One variation of fastball is called two-seam fastball or with-the-seams fastball, where holding the ball with seams visible per rotation increases drag force and thus enhances Magnus Force and bounce. Another type is four-seam fastball or cross-seam fastball.

This text presents information about the four-seam fastball, a type of pitch in baseball. The four-seam fastball has four seams per revolution, resulting in less drag. It can travel up to 2-mph faster than a two-seam fastball and reaches the batter six inches faster. However, it generates a smaller Magnus Force and has less hop compared to the two-seam fastball. The grip for throwing these fastballs is shown in the diagram below, with green lines representing the pitcher’s fingers. Another type of fastball is the split finger fastball, which is thrown by holding the ball like a four-seam fastball but with fingers on the smooth part instead of stitches. This allows for easier slipping off of fingers and reduces rotation, causing an even greater decrease in Magnus Force and making this pitch sink.

Although the spitball violates regulations, its physics can be easily explained. The pitch is thrown with a split finger grip and additional lubricant on the pitcher’s fingers to minimize ball rotation. Consequently, the ball travels rapidly with minimal spin, resembling a fastball. This results in an incredibly unpredictable and devastating pitch, which explains its illegal status.

In contrast, the curveball is extensively studied and considered the most intricate pitch. Physicists have dedicated years of research to comprehend the factors causing a curveball to curve. The origins of the curveball can be traced back to 1671 when Isaac Newton observed a tennis ball curving upon impact. Since then, researchers have been investigating the properties and behavior of the curveball based on Newton’s observations.

Initially, it was thought that the ball’s curve was merely an optical illusion resulting from the pitcher’s spin. However, high-speed photography has proven that a curveball does indeed curve. By capturing images of the ball during its flight to home plate, fast photography has revealed the presence of the Magnus Force, an unbalanced force causing the ball to curve. When the pitcher imparts additional spin on the ball for this pitch, the rotation and stitches create a pressure difference between the ball’s sides. Consequently, one side moves faster than the other, resulting in the ball’s curve. To throw a curveball, the pitcher must apply spin perpendicular to the ground rather than parallel.

In Coors Field, pitchers notice that their curveball does not have as much bite compared to other parks in the league due to the air density being 5,280 feet above sea level in Denver. Another challenging pitch to catch is the knuckleball, known for its unpredictable movement. Pitchers aim to minimize rotation on the ball, causing it to float towards the plate with low initial velocity and high drag force. This large drag force leads to unpredictable movement and late-breaking motion, presenting difficulties for both catchers and batters. Even slight variations in grip or throw can alter the pitch’s trajectory and result in an undesirable path.

The knuckleball can change directions during its flight because the stitch orientation changes due to its lack of rotation. A fastball, on the other hand, maintains a constant stitch orientation because it is thrown with high velocity, causing the ball to rotate consistently. The knuckleball, thrown with low velocity and minimal force, allows for multiple stitch orientations as it travels to the plate. Consequently, the Magnus Force constantly changes directions, leading the ball to move in different directions. Without these directional changes, the ball would be easily hit since it lacks rotation and velocity. Despite some inconsistencies, certain pitchers master the knuckleball technique, making them effective pitchers with an uncontrollable and often devastating pitch.

Physics can analyze various aspects of the game, including hitting. Hitting is a difficult art to master. In Major League Baseball, rules dictate that the bat must be no wider than 2 inches at its thickest part and cannot exceed 42 inches in length. Opinions differ on whether a heavy or light bat is more effective for hitting. A light bat enables a faster swing, allowing the hitter more time to observe the pitch. However, it also reduces the power compared to a heavier bat. According to the equation momentum = mass velocity, a heavy bat can provide more power in the swing. However, if the batter lacks enough velocity in their swing, the use of a heavy bat becomes irrelevant as the gained momentum would be lost.

Taller players need longer and therefore heavier bats compared to shorter players. Scientists have conducted research on the game to find a way to decrease the weight of bats without affecting the hitting surface. Experimental research revealed that the top part of the bat, above the barrel, is where most of the excess weight can be removed while still maintaining the bat’s active area. Players have been able to lighten their bats by sanding down a circular area on the top. As a result, all players can reduce the weight of their bats without compromising the effective hitting surface.

Throughout history, pitchers have been the primary force in baseball, while hitters have aimed to gain an unfair edge against them. This has included employing forbidden substances and methods to tamper with their bats. One particular method, known as corking, entails substituting some of the wood at the center of the bat with cork. This modification results in a stronger impact when hitting the ball and consequently allows it to travel a greater distance. Nevertheless, it is crucial to acknowledge that a bat altered in such manner is less sturdy than a regular one and more prone to breaking.

If a bat splits open, the cork inside will be exposed and the player will face suspension for using illegal equipment. Historically, smaller players who are not known for their hitting abilities have been associated with using corked bats. The most recent player caught using a corked bat had a very poor batting average of .211. Using a corked bat may potentially enhance a batter’s power swing but it usually has either a negative effect or no effect at all. To determine how far a ball will travel, an understanding of the coefficient of restitution is crucial. This physics term describes the effects on both the bat and the ball before and after they make contact.

The coefficient of restitution refers to the amount of energy that remains in the ball after it comes into contact with the bat. A higher coefficient means more energy is retained, resulting in a ball with greater post-impact energy. Conversely, a lower coefficient indicates some absorbed energy during the collision, leading to less distance covered by the ball.

To achieve the highest possible coefficient, batters need to make contact with the sweet spot of the bat. This area allows for greater conservation of energy compared to other parts of the bat, resulting in longer and faster hits. Experienced hitters can easily recognize when they hit this sweet spot because it feels as though they didn’t even touch the ball.

This explains how players like Mark McGwire and Sammy Sosa can anticipate hitting a home run even before the ball clears the fence.

Research also explores how baseball players are able to generate such significant power when hitting the ball. They are taught to transfer their weight from their back foot to their front foot as they swing, allowing their hands and bat to reach speeds of about 70-mph. The speed of the bat is crucial, as even a small difference in timing, such as 0.01 seconds, can determine the outcome of the hit, whether it is a homerun or a pop up, and whether the ball is fair or foul (Adair 50). The placement of the batter’s hands also affects the power generated from the swing. To maximize power, batters can hold the bat at its end. However, this requires more time to get the bat over the plate, increasing the risk of swinging too early if faced with an off-speed pitch.

The equation below demonstrates that increasing the swinging radius leads to higher bat velocity: Velocity of bat = velocity of swing * radius of swing.
Although holding the bat closer to the barrel, also known as choking-up, does not generate as much power as indicated in the equation, it allows for a shorter swing, giving the batter more time to decide if the pitch will be a strike or a ball. Smaller players often choose to choke-up in order to have better control over the bat. Pee Wee Reese and Chuck Knoblauch are examples of players who prioritize hitting for average rather than power. The outcome of a batter’s swing can also be influenced by their technique. The ball typically approaches the batter at an angle of approximately 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 homeruns. They are able to hit the ball with great precision and can usually time the pitch so that they 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. 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. Contrary to Carew and Boggs, Hall of Famers Reggie Jackson and Harmon Killebrew hit the ball with an uppercut swing.

This type of swing typically occurs when the ball is approaching at a 10? angle, while the bat angle is around 25?. While Carew and Boggs focus on hitting line drives, Jackson and other power hitters intend to hit the ball into the air with great force using an uppercut swing. If both types of swings were to hit the ball with equal power, Carew’s ball would land between the outfielders, resulting in a double, whereas Jackson’s ball would end up in the bleachers for a home run at Yankee Stadium. The game of baseball, known as the national pastime, has been extensively analyzed by researchers and scientists. The complex concepts of physics can explain both pitching and hitting strategies. Every aspect of the game, from a curveball to a powerful homerun, has underlying forces that require an in-depth understanding of the physics involved. Mr. Stengel had the correct understanding.

Works Cited

  1. Adair, Robert K. The Physics of Baseball. New York: Harper & Row Publishers, 1990.
  2. Conley, Ken; Chang Choi, and Joe Giuliani. Baseball: The Game and Beyond. Online. Internet. Available WWW: Fitzpatrick, Richard.
  3. The Physics of Baseball Pitching. Online. Internet. Available WWW: League Baseball.
  4. Rules of Baseball. Online. Internet. Available WWW: Monteleone, John and Mark Gola.
  5. The Louisville Slugger Ultimate Book of Hitting. New York: Henry Holt & Company, Incorporated, 1997.
  6. Rubin, Louis D. The Quotable Baseball Fanatic. New York: The Lion Press, 2000.
  7. Sakamoto, Garett. The Physics of Hitting a Baseball. Online. Internet. Available WWW:, Tom. Physics of Sports Lecture 18. Online.
  8. Internet. Available WWW:, Robert G. and A. Terry Bahill. Keep Your Eye on the Ball: The Science and Folklore of Baseball.
  9. New York: W.H. Freeman and Company, 1990.

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