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Analysis of Baseball Pitches
Ling Xu
16-741 Final Project
April 26, 2006
Abstract
Scientists have analyzed the flight of a baseball for more than a century.
Question such as “Does the flight of a baseball curve?” and “Do the seams
on the ball affect its trajectory?” have been explored. The most recent
work has been done by Dr. LeRoy Alaways in defining the fundamental
components of the dynamics of a baseball which help to answer these
questions. His approach will be explained in this paper.
Introduction
In Major League Baseball, pitchers must know a variety of pitches. These
pitches vary elements such as speed and spin in an effort to prevent the batter
from hitting the ball effectively. The finger contacts of the ball as well as arm
and wrist motions allow the pitcher to move the ball in a variety of directions.
A baseball is a leather ball which weighs about 5 ounces. It has a seam
running along the surface of the skin with the same pattern as a tennis ball.
(Fig. 1) This seam plays an important role in the trajectory of the ball by
helping to change the air flow around the ball.
Baseball pitches
There are a variety of pitches in baseball. Their difference lies in distinct finger
contacts, wrist and arm motions. When throwing a fastball or a curveball,
the pitcher lets the ball roll off the fingers. In contrast when throwing the
knuckleball, the pitcher holds the ball with his fingertips and pushes the ball
Figure 1: Baseball [1]
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Figure 2: Magnus Force for a ball without and with spin [3]
into the air. The result is a particular course for the ball as it travels through
the air.
There are also pitches which rely on seam orientation. The “four seam”
fastball requires the placement of the forefinger and middle finger along the
four seams along the ball with the thumb underneath for balance. This pitch
is usually thrown with some backspin and generally goes straighter than most
other pitches with a possible lift in its flight. The “two seam” fastball requires
the placement of the forefinger and middle finger along the two seams along
the ball with the thumb underneath for balance. This pitch generally generally
sinks in flight and sometimes will curve to one side.
Trajectory Analysis
Some important elements affect the trajectory of a baseball. Elements such
as gravitational forces and aerodynamic forces play an important role in the
flight of the ball. One analysis done by Alaways during the late 1990’s tried to
determine the importance of the spin of a baseball to its trajectory.
Magnus Effect
When spin is imparted on a baseball, the resulting trajectory will curve. The
curve in flight of a spherical object was first explained by G. Magnus and the
phenomenon was thus named the Magnus Effect. Magnus said that when a ball
is thrown without any spin, it will break the air evenly along the sides of the
ball creating a wake behind it. In this case there is a backward drag on the
ball as it moves forward. (Fig. 2 top) However if the ball is spinning, the air is
broken differently. The air is separated further along the object in the direction
of the spin and closer to the object in the opposite direction. (Fig. 2 bottom)
Due to this effect to the surrounding air, the trajectory of a spinning baseball
will curve as it travels.
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Aerodynamics
The two main forces acting on a pitch are gravitational and aerodynamic forces.
These forces affect the velocity of the baseball as it travels through the air. The
main equations describing these effect are the following:
FG + FA = m
ΣMG = IG
dV
dt
dω
dt
(1)
(2)
The gravitational forces are defined:
FG = mg
(3)
The aerodynamic force acting on the baseball consists of components such
as lift, drag, and cross force. Lift is the force component acting perpendicularly
to the translational velocity vector while drag is the force component acting
parallel but against the translational velocity vector. The cross force is the
force perpendicular to both lift and drag caused by the seams on the baseball.
The equation for lift is
L=ρ
CL A|V |2 ω × V
2
|ω × V |
(4)
where CL is the lift constant, ρ is the fluid density of the air, A is the
cross-sectional area of the ball, and V is the translational velocity of the ball.
And drag can be characterized by
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D = − ρCD A|V |V
2
where CD is the drag constant
Finally cross force is define as
Y =
1
L×D
ρCY A|V |2
2
|L × D|
(5)
(6)
where CY is the cross force constant
Trajectory of the center of mass
Now that the aerodynamics of the baseball have been characterized, the trajectory of the travel can now be analyzed. First a coordinate system must be
defined. The origin is chosen to be at the center of the ball. The x and y-axis
are defined such that a pure two-seam fast or curveball will be a pure rotation
along either axis. The z-axis is defined to be perpendicular to the x-y axis.
Using rigid body dynamics, the forces of the baseball are defined as the
following:
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mV˙x
mV˙y
mV˙z
= Dx + Lx + Yx
= Dy + Ly + Yy
= Dz + Lz + Yz + FG
Defining the cross product of ω and V :


 V z ω y − ω z Vy 
Vx ωz − ωx Vz
ω×V =


Vy ωx − ω y Vx
Plugging these into the angular momentum equation from (4):
Lx
=
Ly
=
Lz
=
ρCL A|V |(Vz ωy − ωz Vy )
2ω
ρCL A|V |(Vx ωz − ωx Vz )
2ω
ρCL A|V |(Vy ωx − ωy Vx )
2ω
Next we separate the drag into each of the three components:
Dx
Dy
Dz
1
= − ρCD A|V |Vx
2
1
= − ρCD A|V |Vy
2
1
= − ρCD A|V |Vz
2
Then we calculate the cross force:
Yx
=
Yy
=
Yz
=
ρCY A 2
(Vy ωx − Vx Vy ωy − Vx Vz ωz + Vz2 ωx )
2ω
ρCY A 2
(Vz ωy − Vy Vz ωz − Vy Vx ωx + Vx2 ωy )
2ω
ρCY A 2
(Vx ωz − Vz Vx ωx − Vz Vy ωy + Vy2 ωz )
2ω
Finally we can substitute lift, drag, and cross force equations into the acceleration model:
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Figure 3: Lateral deviation of pitches with spin (solid line), measured (circles),
and without spin (dotted line) [4]
V˙x
=
V˙y
=
V˙z
=
ρA|V | CL
Cy 2
[
(Vz ωy − ωz Vy ) − CD Vx +
(V ωx − Vx Vy ωy − Vx Vz ωz + Vz2 ωx )]
2m ω
ωv y
Cy 2
ρA|V | CL
[
(Vx ωz − ωx Vz ) − CD Vy +
(V ωy − Vy Vz ωz − Vy Vx ωx + Vx2 ωy )]
2m ω
ωv x
Cy 2
ρA|V | CL
[
(Vy ωx − ωy Vx ) − CD Vz +
(V ωz − Vz Vx ωx − Vz Vy ωy + Vy2 ωz )] − g
2m ω
ωv x
Results
Alaways conducted tests of over a hundred pitches. He compared the actual trajectories, measured using a high-speed camera, with the estimated trajectories
using the dynamics equations. He found that the average uncertainty between
the positions calculated using the dynamics equations and positions from the
real projection was about 1.4cm. Alaways also calculated the theoretical trajectory of the same ball if thrown with no spin to demonstrate the curvature of
the estimated and real projections.
The trajectories of the pitches with and without spin are shown. (Fig. 3)
The dotted line demonstrates the lateral deviation of a baseball without spin,
the solid line shows the estimated deviation with spin, and the circles are the
measurements from the actual pitch. As shown, the estimated path of the ball is
very close to the measured path. In contrast, the trajectory of the ball with spin
follows a curved path while the path of a spin-free ball has little aberration. The
vertical and horizontal position of the baseball is similar for all three trajectories.
(Fig. 4) This shows the spin causes the ball to shift in the y direction, but not
change position along either the x-axis or z-axis.
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Figure 4: Position of trajectories with spin (solid line), measured (circles) and
without spin (dotted line) [4]
Conclusion
Through multiple field tests using these dynamics equations, Alaways concluded
that when the ball is pitched with a spin, the trajectory of the ball will curve
more than a ball pitched with no spin. He also found that the orientation of the
seam of the baseball is important in the determining the amount of lift the ball
has during flight. For instance “four-seam” pitches tend to remain at a constant
height during flight and “two-seam” pitches tend to fall faster during its travel.
However these differences are mitigated when the spin of the baseball is large.
One additional area to study would be the trajectory of a knuckleball. Because the baseball is held with the fingertips of the hand, the ball is thrown
from the hand with a minimal amount of spin. This small amount of rotation
causes the baseball to wobble as it travels as the seams slowly cut through the
air and create turbulence. It would be interesting to see what forces influence
the trajectory of the knuckleball.
References
[1] http://chesterfield.k12.va.us/Schools/Matoaca HS/images/sports/baseball/ball.jpg
[2] Alaways, L.W. “Aerodynamics of the Curve-Ball: An Investigation of the Effects
of Angular Velocity on Baseball Trajectories.” Doctoral Thesis. University of
California, Davis. 1998.
[3] Kaat, J. “The Mechanics of Breaking Pitch.” Popular Mechanics. 174(10): 52-57.
1997.
[4] Alaways, L.W., Mish, S., and Hubbard, M. “Identification of Release Conditions
and Aerodynamic Forces in Pitched-Baseball Trajectories.” Journal of Applied
Biomechanics. 17: 63-76. 2001
[5] Mason, M. Mechanics of Robotic Manipulation. Massachusetts: MIT Press. 2001.
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[6] Weiss, P. “Pitching Science: Engineers who track baseballs catch insights into the
game.” Science News. 159(23): 366. 2001.
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