Physics of Baseball

"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 ј ounces, and that the circumference of the ball must be between 9 inches and 9 ј- inches (www.majorleaguebaseball.com/library/rules.sml).

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.

(http://farside.ph.utexas.edu/teaching/329/lectures/node79.html)

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 homeruns 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.

(http://farside.ph.utexas.edu/teaching/329/lectures/node79.html)

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 longarmed or shortarmed motion, and you have twenty-four pitches.

-Ed Lopat, 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: