FEATURE ARTICLE
Predicting a Baseball's Path
A batter watches the pitcher's motion plus the spin on the ball to calculate when and where it will cross the plate
A. Terry Bahill, David Baldwin, Jayendran Venkateswaran
Imagine being at the center of the most dramatic moment in baseball.
It's the bottom of the ninth inning of the seventh game of the World
Series—two outs, the tying run on second, the winning run on
first, and you are the batter. Everything depends on you. The
trouble is: The most fearsome pitcher in baseball stands on the
mound. He has an awesome assortment of pitches: fastball, change-up,
curveball, slider and knuckleball. You want any advantage that you
can get in predicting where each pitch will go.
With the crowd going wild and sweat pouring from your every pore,
you have to concentrate on the ball that is about to be launched in
your direction. You must gather as much information about the pitch
as quickly as you can in order to make crucial decisions.
As we will show, you get just a few hundreds of milliseconds to
figure out what kind of pitch—perhaps traveling at almost 100
miles per hour—is heading toward the plate. In that instant,
you must observe the ball's spin and predict how it will move on its
way to the plate. It's a daunting computational task. Luckily, we
can describe a few clues for you to use. And you will need them
soon, because that fearsome pitcher is rocking back on his pivot
leg. In a split second, his arm will swing through a great arc and
send a baseball hurtling your way.
The Physics of a Pitch
Before you have to figure out the World Series–winning or
–losing pitch, let's learn more about the entire process. A
pitcher stands on the mound and throws a baseball—a bit under
three inches in diameter and covered in leather—toward home
plate, which is 60.5 feet away from the pitcher's rubber on the
mound. A strike must cross the plate, which is just 17 inches wide,
at a height that is roughly between a batter's knees and armpits. An
extremely fast pitcher can throw a baseball that reaches 95 miles
per hour, maybe a little faster. At that speed, a ball reaches home
plate in less than half a second. On the way from pitcher to home
plate, though, several forces determine a baseball's trajectory.
As soon as a pitcher releases a ball, it's in gravitational free
fall, whether it's a blistering fastball or a gentle change-up. A
95-mile-per-hour fastball drops 1.7 feet between the pitcher's
release point and the point of a bat-ball collision. Slower pitches
fall more. A 75-mile-per-hour curveball, for instance, drops 5.7
feet. Clearly, a ball's pathway to the plate also depends on other forces.
A pitcher cannot control gravity, but he can put spin on a pitch.
During the nearly two centuries that baseball has been played,
pitchers have invented more than a dozen pitches, and each is
characterized by its specific spin rate, spin direction and forward
velocity. A pitcher controls these characteristics by assuming a
grip and wrist movement devised to provide a given trajectory.
Spin on a ball creates a so-called Magnus force. In the
mid-1850s, German physicist and chemist Gustav Magnus was one of the
first scientists to study this effect. Imagine watching any ball
moving right to left with topspin—meaning that the top of the
ball rotates in the direction of flight. Air flows smoothly around
the ball until it gets to about one o'clock on the top and four
o'clock on the bottom. At those positions, called separation
points, the airflow changes into a turbulent wake that deflects
upward with this spin. The physics behind this force can be
explained in a couple ways. The first invokes Bernoulli's
principle, postulated by 18th-century Swiss mathematician,
Daniel Bernoulli. When a ball with topspin is placed in moving air,
the movement of the ball and its seams slows down the air flowing
over the top of the ball and speeds up the air flowing underneath
it. According to Bernoulli's equation, the point with lower
speed—the top—has higher pressure and the point with
higher speed—the bottom—has lower pressure. This
difference in pressure produces the Magnus force, which pushes the
ball downward. This model has not been validated experimentally.
The second—and probably better—model of the Magnus force
has been validated by wind-tunnel tests. It involves the principle
of conservation of momentum. With topspin, the wake of
turbulent air behind the ball is deflected upward. Anyone can prove
that a body moving in air goes the opposite direction of the
deflected air, which conserves momentum. With a driver aware of your
plan, put your hand out the window of a moving car, and tilt it so
that air is deflected downward; your hand will be pushed upward.
Now, let's relate that to a baseball with topspin moving
horizontally in air. Before the ball interacts with the air, all the
momentum is horizontal. Afterward, the air in the wake has upward
momentum. The principle of conservation of momentum requires that
the ball have downward momentum, which makes it go down.
Of course, a pitcher can put a wide variety of spins on a ball. A
couple of easy "hand" rules reveal which way a spinning
ball will travel. The so-called angular right-hand rule reveals the
spin axis of a pitch. If you curl the fingers of your right hand in
the spin direction, your extended thumb will point in the direction
of the spin axis. For instance, if a ball is spinning in a
counterclockwise direction when viewed from above—as in a
right-handed pitcher's curveball or a left-handed pitcher's
screwball—the thumb will be pointing upward.
Once you know the spin axis, you can find the spin-induced
deflection with the coordinate right-hand rule. Point the thumb of
your right hand in the direction of the spin axis, and point your
index finger in the direction of forward motion of the pitch. Bend
your middle finger so that it is perpendicular to your index finger.
Your middle finger will be pointing in the direction of the
spin-induced deflection. In our example of a pitch with a
counterclockwise spin when viewed from above, your middle finger
will be pointing toward first base.
Get a Grip
A pitcher varies the direction of the deflection by varying the
angle of the spin axis. The spin rate and forward velocity of the
ball determine the magnitude of the deflection. To fine-tune his
abilities, a major-league pitcher practices even when not really
throwing. A pitcher with time to kill tosses a ball into his glove
to practice various grips. The pitcher develops a wrist movement and
a grip that is specific for each of the pitches in his repertoire.
For a fastball, a pitcher snaps the wrist directly forward,
releasing the ball with symmetrical force from the tips of the index
and middle fingers. A fastball delivered with an overhand arm motion
produces backspin. That is, the ball's top surface spins back toward
the pitcher, and the bottom spins forward. The Magnus force will
"lift" such a pitch. More accurately, it decreases the
distance the ball falls due to gravity.
To throw a fastball, a pitcher can grip the ball in different ways,
which are described by the position of the fingers relative to the
ball's seams. Actually, the ball has a single continuous
seam—made up of 108 stitches that hold together two smooth
pieces of leather—but this seam curves to fit the surface of a
sphere. If a pitcher grips the ball across the seams, it appears
that four seams pass in front as the ball makes one revolution.
Hence, this is called a "four-seam" grip. If a pitcher
grips a fastball with the seams, it's called a "two-seam"
grip because only two seams appear on the front during a revolution.
Most pitching coaches recommend a four-seam grip for the fastball.
They presume that a seam perpendicular to the trajectory of the
pitch encounters greater air resistance than the smooth surface of
the ball. Therefore, they speculate that a four-seam fastball
encounters greater air resistance than a two-seam fastball, which
might create a stronger Magnus force on the ball. Pitchers assume
this produces a greater lift on the overhand fastball. Indeed,
pitchers have written that the four-seam grip is more effective than
the two-seam grip in producing rising fastballs. However,
wind-tunnel tests have shown no significant differences in lift
between two- and four-seam orientations. Two of us (Baldwin and
Bahill) have explained that the perceived rise of the four-seam
fastball is probably a perceptual illusion.
A fastball, though, can experience more than an upward deflection.
Any delivery that varies from directly overhand will create some
spin at an angle to horizontal, which generates some lateral
deflection. Moreover, sidearm or "submarine" fastballs
tend to have some topspin so these pitches sink more than they would
due to gravity alone.
A Catalog of Curvatures
Even more options arise when it comes to a curveball. First, a
pitcher grips the ball with his middle finger lined up along, or
just inside, one of the seams where the leather makes a roughly
circular shape on the surface of the ball, and his index finger lies
right beside the middle one. In general, a pitcher rotates his wrist
as the ball is released to throw a curveball. This causes four seams
to appear per revolution if you could watch the ball from directly
in front as it heads toward the plate; thus this pitch is also
generated by a four-seam grip. The index and middle fingers roll to
the front or side of the ball, imparting greater spin. An overhand
curve produces topspin, resulting in a downward deflection usually
referred to as a "drop." In other words, a ball drops more
than it would due to gravity alone. If a pitcher applies spin with a
vertical axis, as on a toy top, the ball curves horizontally, and
concurrently falls due to gravity. This "flat" curve is
thrown by pitchers using a sidearm delivery.
Most pitchers adopt what is called a "three-quarter
delivery," swinging the arm through an arc that is roughly
halfway between vertical and horizontal. This applies sidespin and
topspin components to a curve. For a right-handed pitcher, the ball
curves diagonally from upper right to lower left. The speed of a
curveball varies from around 70 to 80 miles per hour, and the spin
rate has been measured at up to 2,000 revolutions per minute.
The drop in a curve usually gives a hitter more trouble than the
sideways deflection, because of the shape of the bat and the
horizontal orientation of the swing. A bat's sweet area—the
place that can hit a ball most effectively—is about 4 inches
long but only one-third of an inch high. As a result, a vertical
drop is more effective than a horizontal deflection at taking the
ball away from the bat's sweet area, because the batter has a
smaller margin of error vertically. On the other hand, a horizontal
curve can be just as hard to hit as a dropping one when thrown to a
batter of like handedness—that is, right-hander to
right-hander or left-hander to left-hander. Anyone who has stood in
the batter's box—even facing a good high school
pitcher—soon learns that it is easier to hit a curve that is
deflected toward you instead of one bending away. Apparently, the
batter finds it harder to judge the horizontal location of the pitch
as it curves away. Also, a batter tends to flinch a bit from a
curveball that is aimed at him and then "breaks" toward
the outside corner of the plate.
Another horizontally deflected pitch is called a "slider."
It travels faster than a curveball, but spins less and,
consequently, only deflects about half as much as an ordinary
curveball. A pitcher throws a slider somewhat like a pass in
football. He takes a fastball grip and rolls his wrist slightly
during the delivery. That makes a pitcher's finger pass toward the
outside of the ball—sometimes called "cutting the
ball"—and that creates some lateral spin. A slider's axis
of rotation usually points up and to the left from the perspective
of a right-handed pitcher. This causes the ball to drop a little and
curve from the right to the left.
The curveball and slider bend away from the pitcher's throwing-arm
side, whereas a screwball deflects the other way. That is, if a
right-handed pitcher throws a screwball, it curves toward a
right-handed batter. In the early 1900s, Christy Mathewson—a
longtime New York Giant and member of the Hall of
Fame—developed this fadeaway pitch. One of Mathewson's biggest
rivals, Hall of Famer Mordecai "Three-Fingered" Brown,
threw a natural screwball because he lost part of his index finger
on his pitching hand in a farm-machine accident when he was a child.
In the 1930s, New York Giant Carl Hubbell made the screwball popular
once more. Of the pitches described in this article, pitchers throw
the screwball the least. It requires difficult twisting of the hand,
forearm and elbow that puts the pitcher's fingers on the inside and
top of the ball at the point of release.
No matter which way a curving pitch goes, once it starts to move, a
batter can predict the trajectory with some confidence. Not so with
a knuckleball. A hurler grips this pitch with his knuckles on a
smooth part of the ball or his fingertips dug into a seam. Then, he
holds his wrist rigid, basically pushing the ball, which reduces any
spin. This pitch travels slowly—only about 60 miles per
hour—and a good one revolves less than one time on its way to
the plate. That spin is unpredictable, as is the ball's trajectory.
For example, Hoyt Wilhelm—a Hall of Famer who was one of the
greatest knuckleballers—threw a knuckler that was described as
following a corkscrew path, attaining multiple deflections during
its flight.
Tricking a batter, though, takes more than throwing the right
curves. Changing the speed of pitches also plays a large role. Even
if a pitcher could throw 100-mile-per-hour fastballs for nine
innings, major league hitters would time the pitches and turn
potential strikes into home runs. So, pitchers also use a
change-up—just an off-speed straight pitch—that can be
thrown in several ways. One of the most common change-up techniques
is the palmball, which is thrown by shoving the ball into the palm
of the pitcher's hand. This reduces the whipping action of the
wrist, and even with a fastball delivery the velocity of the pitch
drops to 60 or 70 miles per hour.
Keep Your Eye on the Ball
As a pitcher delivers a ball, a batter gets a few clues for
developing a mental model of the pitch. For example, the angle of
the pitching arm provides vital information about the upcoming
trajectory of the pitch. Arm angle varies through a continuum that
includes overhand, three-quarters, sidearm and submarine.
Consequently, the height of the release point varies from over six
feet off the ground to just one foot. The release point also varies
for different pitches.
Another clue for the ball's impending behavior is the launch angle.
To go through the strike zone, a 95-mile-per-hour fastball must be
launched downward at a 2-degree angle, whereas a 60-mile-per-hour
change-up must be launched upward at a 2-degree angle. A major
league batter can distinguish the difference between these angles.
An good major-league batter might even be able to distinguish the
difference in launch angle between a fastball and a curveball.
A batter can also look for how the pitcher holds the ball as he
releases it. With the knuckler, a batter will see two or three
knuckles sticking up above the ball as a pitcher releases it. If a
pitcher throws a curveball and a batter has keen eyesight, he might
be able to see the index and middle fingers roll across the face of
the ball as the pitcher snaps it off. These are examples of
information about the kind of pitch that will be coming a batter's way.
A batter's best source of information, however, is the way the ball
is spinning immediately after its release. How much spin a batter
can distinguish, though, depends on his dynamic visual acuity, which
is the ability to perceive moving objects. (An optometrist, on the
other hand, measures a person's static visual acuity, which is the
ability to perceive information in nonmoving objects, such as
letters on a page. Moreover, a person's static visual acuity is not
correlated with his dynamic visual acuity.) A batter needs excellent
dynamic visual acuity to track and predict the flight of a baseball.
Experienced athletes have better than average dynamic visual acuity,
partly because athletes are selected for this ability and partly
because it can be improved with training. Nonetheless, our survey of
major-league hitters revealed considerable variation in their
ability to see the spin on a pitch. Batters with good dynamic visual
acuity can see the spin on the ball; those with poor dynamic visual
acuity cannot. To get a feel for the range in dynamic visual acuity,
consider that most of us can read the label on a phonograph record
turning at 33 revolutions per minute, but this would be about the
limit of our capabilities. The great Boston Red Sox hitter, Ted
Williams, could read one turning at 78 revolutions per minutes,
which is far beyond the dynamic visual acuity of the average person.
And Here's the Pitch …
At last, the moment of truth arrives. The pitcher launches his body
down the mound at you, and his arm suddenly whips out from behind
him. And there is his hand releasing the pitch. The ball is in the
air, spinning and telling you everything you need to know.
As you brace for the pitch, you must evaluate it
quickly—extremely quickly. You must determine its speed and
spin in about one-seventh of a second. In the next one-seventh of a
second, you decide whether to swing and—if you decide
yes—where and when to swing. That leaves just one-seventh of a
second—if the pitch is a fastball—to swing the bat.
In the rest of this article, we will concentrate on the first
roughly 150 milliseconds (about one-seventh of a second) after the
release. In that time, a batter tries to determine the direction and
rate of spin to predict the magnitude and direction of a ball's
deflection. The appearance of the pitch, however, depends on a
pitcher's grip. For example, a two-seam fastball does not look like
a four-seam fastball, although the speed and spin rates of these
pitches are the same.
To prove this, we skewered baseballs on bolts in the four- and
two-seam orientations. The bolts were chucked in electric drills and
rotated at 1,200 revolutions per minute—the typical spin rate
for a fastball—which was measured with a stroboscope. Both
visual observation and photographs show that a four-seam fastball
appears to be a gray blur with thin vertical red lines about
one-seventh of an inch apart and running perpendicular to the spin
axis. These lines are the individual stitches of the baseball, but
even Ted Williams could not see the individual stitches. By
comparison, a two-seam fastball looks different. It exhibits two big
red stripes, each about three-eighths of an inch wide, which are
created by the spinning seams. (For simplicity, our drill-driven
fastballs modeled an overhand delivery. The more common
three-quarter arm delivery would tilt the axis of rotation by 45
degrees.) Those stripes provide easily perceived information for the
batter to determine the angle of the spin and then predict the
direction of the resulting deflection. Therefore, the big difference
between the two- and four-seam fastballs is that—because of
the visibility of vertical red stripes—the batter might be
able to more quickly and easily perceive the spin direction on the
two-seam version.
A video of drills spinning four- and two-seam fastballs shows the
difference even more dramatically. Moreover, the difference in
appearance of the four- and two-seam orientations is even more
apparent for spinning baseballs in our laboratory than it is in this
video. Those differences made us think that something in addition to
the big red stripes distinguishes the two- from the four-seam
pitches. We hypothesized that the difference might relate to the
critical flicker-fusion frequency.
Seeing and Concealing
As the frequency of a blinking light increases, the light appears to
flicker and then, at a certain frequency, it appears to be
continuously illuminated. This transition point is the critical
flicker-fusion frequency, which is measured in hertz (Hz), where 1
Hz equals one pulse per second. For a person in a baseball park,
this frequency is probably between 40 and 50 Hz. Television screens
present a different frame 60 times per second, or 60 Hz, and the
pictures do not flicker. The time indicator on your VCR, on the
other hand, probably blinks once a second, which clearly produces
what is perceived as a blinking sequence. (At the beginning of the
20th century, movies were called "the flicks," because the
24 Hz–frame rate produced flickering images.)
A typical major-league fastball completes about 1,200 revolutions
per minute, or 20 per second. For a two-seam fastball, the pair of
seams that straddle the narrow isthmus of the ball would cross the
field of view once on each rotation. These seams lie so close
together that they probably appear as a single item. Therefore, the
frequency of this pulse would be around 20 Hz, which is below the
critical flicker-fusion frequency, and perhaps the ball would appear
to flicker, giving the batter a clue about the spin. A batter might
not have to compute the spin rate to determine whether the pitch is
a fastball or curveball. Instead, he just has to determine from the
flickering if the ball has topspin—a curveball—or
backspin—a fastball. That could help a batter quickly predict
the movement of a ball.
In contrast, each of the seams of a four-seam fastball would cross
the field of view once per rotation. That produces a frequency of 80
Hz, which is above the critical flicker-fusion frequency. Therefore,
a ball would not flicker, and a batter would not have this extra
clue about spin. A batter would have to guess if a pitch were a
curveball or a fastball. If he guesses curveball when it is really a
fastball, he will expect a pitch to be slower than it actually is.
Therefore, he will also expect the ball to fall farther than it
actually will. Consequently, when he discovers that the pitch is
higher than he predicted, he might perceive a rising fastball. This
could be the reason that pitchers often say the four-seam fastball rises.
One of us (Venkateswaran) measured how far away a non-athlete with
ordinary vision could see the stripes of a simulated two-seam
fastball compared with the thin red lines on a four-seam fastball.
The two-seam stripes showed up at roughly 16 feet versus 10 feet for
the thin lines on a four-seam fastball. Professional baseball
players undoubtedly have better dynamic visual acuity than this
subject and can probably see the red stripes much farther away. For
a professional fastball, the batter's swing starts when the ball is
about 19 feet from the plate. Information gathered after this point
would be of no help for that pitch.
The two-seam versus four-seam grip might also give the batter clues
about the slider. For example, we surveyed 15 former major-league
hitters about what they remembered about a slider. Eight remembered
seeing a dot in the upper-right quadrant of the ball for a slider
thrown by a right-handed pitcher. They remembered seeing this
telltale sign that a pitch was a slider.
To study this, we went back to our drills. We bolted one ball to
spin the way it would for a slider from a four-seam grip and another
to spin as it would from a two-seam grip. The four-seam grip used
for a slider causes the axis of rotation to exit the ball through a
seam, which creates the perception of a red dot. With a two-seam
grip, the axis of rotation exits the ball through an open patch of
white leather, which eliminates a red dot. Generally, pitchers use
the same grip for the fastball and slider to avoid tipping off the
pitch, so using a four-seam grip works to the pitcher's advantage on
a fastball, but presents a distinguishing feature on a slider.
The grip employed for a knuckleball reduces the spin rate, and the
grip used for the palmball reduces the forward velocity of the ball.
A knuckleball baffles a batter because of the ball's erratic
behavior. Even though a batter might see the knuckleball grip as the
pitcher releases the ball, this information will not help a batter
much. The palmball has the same spin axis as the fastball but it has
a slower spin rate and might be spotted quickly.
Physical tests show negligible differences in deflection magnitude
between the two- and four-seam fastballs, curveballs or sliders. The
big differences seem to be psychological—specifically
perceptual. The batter can see the two red stripes and the flicker
of the two-seam fastball and palmball, the two red stripes of the
two-seam curveball and the red dot on a four-seam slider. All of
these clues alert the batter to the type of spin on the ball and
help him predict its movement.
In conclusion, the pitcher should use a four-seam grip for fastballs
and curveballs to remove the perceptual clue of the two red stripes
and the flicker. Then, he should use the two-seam grip for the
slider, to remove the clue of the red dot. These techniques could
make a fearsome pitcher even more difficult to hit. But if you're in
luck, he hasn't read this article.
Bibliography
- Adair, R. K. 2002. The Physics of Baseball. New
York: HarperCollins.
- Bahill, A. T., and
D. G. Baldwin. 2004. The rising fastball and the perceptual
illusions of batters. In Biomedical Engineering Principles
in Sports, ed. G. Hung and J. Pallis. New York: Kluwer
Academic.
- Bahill, A. T., and T. LaRitz. 1984.
Why can't batters keep their eyes on the ball? American
Scientist 72:249-253.
- Bahill, A. T.
Baseball video. http://sie.arizona.edu/sysengr/baseball/2Seam-4Seam-Video.AVI
- Baldwin, D. G., and A. T. Bahill. 2004. A model of the
bat's vertical sweetness gradient. In Proceedings of the 5th
Conference of Engineering of Sport, ed. M. Hubbard, R.
D. Mehta and J. M. Pallis. Sheffield, UK: International Sports
Engineering Association.
- Nathan, A. L.
personal Web site. http://www.physics.uiuc.edu/People/Faculty/profiles/Nathan
- Selin, C. 1959. An analysis of the aerodynamics of
pitched baseballs. The Research Quarterly
30(2):232-240.
- Watts, R. G., and A. T.
Bahill. 2000. Keep Your Eye on the Ball: Curve Balls,
Knuckleballs and Fallacies of Baseball. New York: W. H.
Freeman.