Adapted from the book by Robert Kemp Adair, Sterling Professor of Physics,

Yale University and Physicist to the National League 1987-89.

Illustrations and text adapted from "The Physics of Baseball" published by Harper & Row. Copyright 1990

by Robert K. Adair.

A. A pitcher throwing a major-league fast ball -- which leaves his hand with an initial velocity of 97 mph and crosses the plate 0.4 seconds later at 90 mph -- transmits an average of about 1.5 horsepower to the ball. Energy stored in the stretching of the arm's tendons is transferred to the ball in the last portion of the throw as the spring-energy of the stretched tendons is released. When age or injury reduces the tendons' elasticity, the arm goes "dead."

B. The fluid dynamics of air flow about spheres governs the flight of the baseball. For velocities below about 50 mph, the flow is rather smooth, though trailing vortices are generated. This air flow does not actually reach the surface of the ball, where there is a quiet boundary layer. At velocities above 200 mph, the flow penetrates the boundary layer, and the air at the boundary is quite turbulent. Much of the subtlety of baseball is derived from the fact that so much of the game is played in the region between definitely smooth flow and definitely turbulent flow. By and large, turbulence will be induced at lower velocities by roughness in the surface. Furthermore at a given velocity, the air resistance is, surprisingly, smaller for turbulent flow than for smooth flow. If the baseball were quite smooth rather than provided with 216 raised cotton stiches, it could not be thrown or batted nearly as far: A stitched ball batted 400 feet could travel only about 300 if it were very smooth.As the baseball travels from pitcher to batter, the total drag force on the ball (from the normal air pressure of 14.6 pounds per square inch) pushing the ball toward third base is nearly 100 pounds. Of course, there is, ordinarily, a nearly identical drag force pushing the ball toward first base. If these forces differ by as much as 1.5 oz. -- or about one part in a thousand -- the ball thrown to the plate at a velocity of 75 mph will be deflected, or curve, a little more than a foot. Such modest imbalances are generated by asymmetric spinning of the ball and by asymmetric placement of the stitches.

C. An unbalanced force on the spinning ball occurs because the velocity through the air on one side of the spin equator is greater than the velocity on the other side. Such a force, directed at right angles to the direction of the air velocity, is called the Magnus force. If a curve ball is thrown by a right-handed patcher at 70 mph so that it rotates 17 times counterclockwise (seen from above) in its trip of about 60 feet from the pitcher's hand to the plate, such a ball will be rotating at about 1800 rpm.

The side of the ball toward third base (bottom) then travels about 15 X 9 in. (the circumference of the ball) -- or 11 feet farther than 60 feet. The side toward first base travels 11 feet less. The velocity of the third-base side is about 82 mph, and only 58 mph on the first-base side. We can then expect the air pressure on the third-base side of the ball (which is traveling faster) to be greater than the pressure on the slower first-base side. The ball is deflected toward first base.The figures below show the trajectories of a curve and knuckle ball.

D. Very large forces, reaching values as high as 8,000 lbs., are required to change the motion of the 5 1/8-oz. ball from as speed of 90 mph toward the plate to a speed of 110 mph toward center field in the 1/1000th of a second during which the bat contacts the ball. The ball is compressed to about one-half its original diameter; the bat about 1/50th as much. The figures to the right illustrate the maximum distortion of the ball at various velocities.The ball may be considered as a spring. The bat applies force to the ball, compressing it, and the ball exerts force on the bat on regaining its original contours. The recoil from this exerted force propels the ball away from the bat. The outcome depends on the inelasticity of both ball and bat.The inelasticity is usually described in terms of the coefficient of restitution (COR), which is the ratio of the velocity of the ball rebounding from the surface of a hard, immovable object and the incident velocity. For baseballs traveling 85 feet/second (58 mph), striking a wall of ash boards backed by concrete, the mean COR of a large set of 1985 and 1987 official major league baseballs has been measured at 0.563. That is, the balls rebound with a velocity of 0.563 X 85 ft/sec, or 48 ft/sec. As the collision velocity increases, the COR for baseballs probably decreases.Typically, the fast ball struck by the bat carries about one-fifth as much kinetic energy as the bat. If the ball is struck squarely, about half the energy of the swinging bat is transferred to the ball in the impact, so the speed of the bat is sharply reduced -- by about 30 percent. About one third of the original bat-and-ball energy is carried off as kinetic energy in the flight of the ball from the bat, and the rest of the energy is lost in friction in the course of the distortion of the ball -- and then to heating the ball.

E. The elasticity of balls stored under extremes of heat or cold can be affected. There are apocryphal stories of home-team managers who stored balls on ice before they were given to the umpires when the visiting team was at bat. The balls were taken off the ice a few hours before game time so the covers could warm up and not alert the umpire; but the core would remain cold and dead. Some simple measurements suggest that deep-freezing to -10 F would seem to take about 25 feet off a nominal 375-foot fly ball. Conversely, storing balls in a warming oven at 150 F would seem to inject enough rabbit in the ball to take a 375-foot fly over the fence to about 400 feet from home plate.

F. Assuming that as batter's "time of decision" is typically 0.15 seconds before the ball crosses the plate, a player will gain no more than 0.0133 seconds by changing from a 38-oz. bat to one 6 oz. lighter. For a 90-mph fastball, the batter would gain about 15 inches. For some players, the small gain in time (a quicker swing means an increased likelihood of striking the ball) may be more important than the small loss in speed and flight distance of the hit.

G. The total kinetic energy of man and bat together is constant for different bat weights, but it is divided differently. More of the energy goes into a heavy bat, though the bat -- and player -- velocity is smaller; less energy goes into a light bat, though the velocity is somewhat greater. A mathematical model suggests that a 46-oz. bat propels the ball only 8 feet farther than the 32-oz. bat; and if a player drills a hole in his 32-oz. bat (perhaps filling it with cork) so the weight is reduced to 28 oz., the ball will go only 373 feet. Babe Ruth used 47- and 50-oz. bats. Modern players use 31 to 36 oz.

H. Different kinds of pitches can be generated by employing different patterns of spin, turbulence and dynamic forces. The diagrams to the right indicate the spin direction (as seen by the batter) of various throws by a right-handed pitcher. The direction of rotation is also the direction of the Magnus force. Thus the fastball will "hop" owing to the upward direction of the Magnus force, and so forth. Since a faster ball reaches the plate sooner, the force has a shorter time to act; for the same transverse force, the ball that travels 10 percent faster will curve 20 percent less.The thrown ball can also be deflected by turbulence induced by the stitching. If the ball is thrown with very little rotation, asymmetric stitch configurations can be generated that lead to large imbalances of forces and extraordinary excursions in trajectory. Measurements and wind tunnel simulations have shown that such a pitch -- thrown originally toward the center of the plate -- will be 11 inches off-center when it is 20 feet from the plate. In general, a knuckle ball pitcher tries to throw the ball so that it rotates only half a revolution over the entire distance to the plate, so that the stitch configurations -- and forces on the ball -- change on the way. Such a slowly rotating ball can be thrown off the knuckles (with the ball held between the forefinger and little finger), but in practice is usually thrown off the fingertips or fingernails. I and J. Does a curve ball really curve in a smooth arc? Or does it "break" as it nears the plate? The illustration (top right) shows a wide-breaking curve thrown by a right-handed pitcher. This ball is thrown with an initial velocity of 70 mph, spinning at 1600 rpm, to cross the plate about 0.6 seconds later at about 61 mph. Although the radius of the curvature is nearly constant throughout the ball's flight, the deflection from the original direction increases approximately quadratically with the distance -- i.e, four times the deflection at twice the distance.Halfway from the pitcher to the plate, the ball has moved about 3.4 inches from the original line of flight, which is directed toward the inside corner and is moving toward the center of the plate. At the plate, the deflection is 14.4 inches and the ball passes over the outside corner. One half of the deflection occurs during the last 15 feet. Thus neither the smooth arc nor the break is an illusion: Each is a different description of the same reality. K. and M. The diagram above shows that a ball batted with an initial velocity of 110 mph at an angle of 35

from the horizontal would go about 750 feet in a vacuum; at Shea Stadium in New York, it will travel only about 400 feet.The illustration below shows force versus compression distance for various simulated bat-ball collisions and for a golf ball. The area under the upper curve is proportional to the energy absorbed by the ball in motion; the area under the lower curve is the energy returned by the ball in its resumption of its spherical shape. The area enclosed by the two curves is proportional to the energy dissipated or lost in friction. For an ideal spring, the two curves will coincide. As shown here, the baseball returns only about 35 percent of energy supplied in compression. The golf ball, much closer to a perfect spring, returns more than 75 percent of the compressive energy when struck by a driver. N. Given a bat 35 inches long, the maximum energy transfer from the stiff bat to the ball occurs when the ball is struck at a point about 30 inches from the handle. But bats vibrate when the ball is hit too far from the optimal point (that is, a "vibrational node" or point of no vibration), resulting in a weakly hit ball and often a broken bat. The diagram (right) shows distortions of a bat when striking the ball near a node and an antinode respectively. For a typical bat, the oscillation frequency is about 260 cycles per second. At that frequency, the half-cycle time of about 0.002 seconds is appreciably longer than the natural bat-ball interaction time of about 0.0005 seconds. Thus the bat does not return the energy of distortion to the ball, but retains that energy in the vibration familiar to any baseball player. O. For a hard-hit ball traveling with an initial velocity of 110 mph, the spin rate will decrease at a rate of about 30 percent per second. For a 400-foot home run, the backspin applied by the bat (perhaps 2,000 rpm) is reduced to about 350 rpm when the ball lands about 5 seconds later. Maximum distance is obtained at an initial angle of 35

to the horizontal. A 385-foot fly ball hit at 35

will be in the air for about 5 seconds. (An average right-handed batter will run from home to first in about 4.3 seconds.) The maximum distance any ball can be hit is probably about 550 feet.