How a Physicist Analyzes the
Game of Baseball
Alan M. Nathan
a-nathan@uiuc.edu
webusers.npl.uiuc.edu/~a-nathan/pob
Department of Physics
University of Illinois
1
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
1927
Solvay Conference:
Greatest physics team
ever assembled
MVP’s
2
A great book to read….
“Our goal is not to reform the
game but to understand it.
“The physicist’s model of the
game must fit the game.”
3
Some Topics I Will Cover
• How does a baseball bat work?
• The flight of a baseball
• Leaving the no-spin zone
• Putting it all together
4
“You can observe a lot by watching”
--Yogi Berra
Champaign News-Gazette
Easton Sports
5
CE Composites
Description of Ball-Bat Collision
• forces large, time short
– >8000 lbs, <1 ms
• ball compresses, stops, expands
– KEPEKE
– bat recoils
• lots of energy dissipated (“COR”)
– distortion of ball
– vibrations in bat
• to hit home run….
– large batted ball speed
• 100 mph~400 ft, each additional mph ~ 5-6’
– optimum take-off angle (300-350)
– lots of backspin
6
Kinematics of Ball-Bat Collision
vball vbat
BBS = q vball + (1+q) vbat
BBS
• q  “Collision Efficiency”
• Joint property of ball & bat
 independent of reference frame
 ~independent of “end conditions”—more later
 weakly dependent on vrel
• Superball-wall: q  1
• Ball-Bat near “sweet spot”: q  0.2
 BBS  0.2 vball + 1.2 vbat
Conclusion:
vbat matters much more than vball 7
Kinematics of Ball-Bat Collision
BBS = q vball + (1+q) vbat
e-r
q=
1+r
 e-r 
1+e 
BBS = 
v ball  
v bat


1+r 
 1+r 
vball vbat
BBS
e: “coefficient of restitution”  0.50
(energy dissipation—mainly in ball, some in bat)
q=0.20
r = mball /Mbat,eff : bat recoil factor =  0.25
(momentum and angular momentum conservation)
---heavier is better but…
8
Batting cage study show how bat speed depends
on MOI for college/semipro baseball players
Crisco/Greenwald Batting Cage Study
50
y = m1*(9/m0)^m2
Value
Error
m1
46.218
0.3921
m2
Chisq
0.28747
3.8574
0.057422
NA
R
0.93138
NA
48

knob
46
(rad/s)
44
aluminum
wood
42
40
8.5
9
9.5
10
3
10.5
2
I (10 oz-in )
6"
11
11.5
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Collision Efficiency q Can Be Measured
• Air cannon fires ball onto stationary bat
• q = vout/vin
• Used by NCAA, ASA, … to regulate/limit
performance of bats
Sports Sciences Lab @ WSU
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Regulating Performance of NonWood Bats (NCAA)
BBS = q vball + (1+q) vbat
• Specify maximum q (“BESR”=q+1/2)
– relative to wood
– implies bats swung alike will perform alike
• Specify minimum MOI to limit bat speed
– smaller than wood
• Together, these determine a maximum
BBS
– gap between wood and aluminum  5%
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HR
NCAA home runs/game
1.15
1.05
0.95
0.85
0.75
0.65
0.55
0.45
0.35
70 972 974 976 978 980 982 984 986 988 990 992 994 996 998 000 002 004 006
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1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
year
aluminum
-5 rule
BESR MOI
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Accounting for COR:
Dynamic Model for Ball-Bat Collision
AMN, Am. J. Phys, 68, 979 (2000)
• Collision excites bending vibrations in bat
– hurts! breaks bats
– dissipates energy
• lower COR, BBS
• Dynamic model of collision
– Treat bat as nonuniform beam
– Treat ball as damped spring
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Modal Analysis of a Baseball Bat
www.kettering.edu/~drussell/bats.html
f1 = 179 Hz
f3 = 1181 Hz
f2 = 582 Hz
f4 = 1830 Hz
FFT(R)
1
0.15
582
0.5
0
R
5
10
15
20
25
30
1181
-0.5
-1
1830
179
0.05
-1.5
35
0.1
0
frequency
time
0
5
10
t (ms)
15
2400
20
0
0
500
1000
1500
frequency (Hz)
2000
14
2500
Vibrations, COR, and the “Sweet Spot”
Strike bat here
nodes
4 3 2
0.5
0.4
e
v (mph)
120
best performance & feel
100
e
0.4
+
f
1
vf
0.3
80
@ ~ node 2
60
0.3
0.2
Evib
0.2
0.1
0
5
10
distance from tip (inches)
40
20
0
15
15
Independence of End Conditions
• strike bat in barrel—look at response in
handle
• handle moves only after ~0.6 ms delay
• collision nearly over by then
v (m/s)
• nothing on knob end matters
• size, shape
• boundary conditions
• hands!
• confirmed experimentally
30.00
20.00
10.00
0.00
-10.00
-20.00
-30.00
0
1
2
3
t (ms)
4
165
q independent of end conditions:
experimental proof
Collision Efficiency
0.25
0.2
0.15
"normal" bat
0.1 normal + 3 oz in knob
0.05
0
25
26
27
28
29
30
31
distance from knob (inches)
32
Conclusion: mass added in knob has no
effect on collision efficiency (q)
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Vibrations and Broken Bats
pitcher
movie
0.000
5.000
10.000
15.000
20.000
25.000
30.000
35.000
catcher
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Why Is Aluminum Better Than Wood?
Aluminum has thin shell
– Less mass in barrel
--lower MOI, higher bat speed, easier to control 
--but less effective at transferring energy 
--for many bats  cancels 
» just like corked wood bat
– “Hoop modes”
• trampoline effect  
• “ping”
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demo
The “Trampoline” Effect:
A Simple Physical Picture
•Two springs mutually compress each other
KE  PE  KE
• PE shared between “ball spring” and “bat spring”
• PE in ball mostly dissipated (~80%!)
• PE in bat mostly restored
• Net effect: less overall energy dissipated
...and therefore higher ball-bat COR
…more “bounce”—confirmed by experiment
…and higher BBS
• Also seen in golf, tennis, …demo
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Aerodynamics of a Baseball
• Gravity
• Drag (“air resistance”)
• Lift (or “Magnus”)
FM
v

ω
Fd
mg
Fd=½ CDAv2
-v direction
FM = ½ CLAv2
(ω  v) direction
Courtesy, Popular Mechanics
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direction leading edge is turning
Measuring drag and Magnus forces
by high-speed tracking
drag/wt0.8 Magnus/wt=0.58
1540
3000
y
1520
2000
1500
1000
1480
0
y
-1000
-2000
-3000
0.00
1460
z
1440
<v>=71.5 mph
=4989 rpm
z
Magnus force
is much easier
to measure
than drag force
1420
1400
0.04
0.08
0.12
0.16
t (s)
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Typical values of drag and lift
2
“Drag crisis?”
1.5
Drag/Weight
1
0.5
0
Lift/Weight
@1800 rpm
0
25
50
75
100
Speed in mph
125
150
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Effect of Drag and Lift on
Trajectories
FM
120
v

ω
100
no drag or lift
80
60
Fd
40
mg
drag, no lift
drag and lift
20
0
0
100
200
300
400
500
600
700
distance (ft)
• drag effect is huge
• lift effect is smaller but significant
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Some Effects of Drag
120
• Reduced distance on fly ball
• Reduction of pitched ball
speed by ~10%
100
no drag or lift
80
60
40
drag, no lift
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• Asymmetric trajectory:
– Total Distance  1.7 x distance
at apex
• Optimum home run angle
~30o-35o
0
0
100
200
300
400
500
600
700
80
90
distance (ft)
Range (ft)
400
2000 rpm
350
300
0 rpm
250
200
150
100
Range vs. 
50
0
10
20
30
40
50 60
 (deg)
70
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Some Effects of Magnus
FM
• Backspin makes ball rise

– “hop” of fastball
– undercut balls: increased distance,
reduced optimum angle of home run
• Topspin makes ball drop
– “12-6” curveball
– topped balls nose-dive
• Breaking pitches due to spin
– Cutters, sliders, etc.
v
ω
Fd
mg
120
100
no drag or lift
80
60
40
drag, no lift
drag and lift
20
0
0
100
200
300
400
distance (ft)
500
600
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700
The PITCHf/x Tracking System
A New Tool to Study Baseball Flight
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How Does PITCHf/x Work?
• Two video cameras track baseball in 1/60-sec intervals
– usually “high home” and “high first”
– third CF camera used establises ht. of strike zone for each batter
• Pattern-recognition software to identify “blobs”
• Camera calibration to convert pixels to (x,y,z) 9parameter fit to trajectory
– constant acceleration for x(t),y(t),z(t)
• Use fit to calculate lots of stuff
– The full trajectory
– The “break”
– Drag and Magnus forces
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Pitch Classification
Jon Lester, Aug 3, 2007 @ Seattle
LHP
Catcher’s View
I: Nearly overhand fastball
II: Cut Fastball
III: ¾ Fastball
IV: Curveball
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What’s the Deal with the Gyroball?
Courtesy, Ryutaro Himeno
Courtesy, The New York TImes
Daisuke Matsuzaka:
Does he or doesn’t he?
30
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Barry Bond’s 756th Home Run
• PITCHf/x data tracked hit ball over first 20 ft
• Precision measurement of endpoint and time
• Inferred: v0=112 mph; =27o up; =16o to right of
dead center; =1186 rpm (backspin) and 189 rpm
(sidespin, breaking to center)
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Baseball Aerodynamics:
Things I would like to know better
• Better data on drag
– “drag crisis”?
– Spin-dependent drag?
– Drag for v>100 mph
• Dependence of drag/lift on seam orientation
• Is the spin constant?
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Oblique Collisions:
Leaving the No-Spin Zone
Oblique  friction  spin
Familiar Results:
• Balls hit to left/right break toward foul line
• Topspin gives tricky bounces in infield
• Backspin keeps fly ball in air longer
• Tricky popups to infield
demo
34
Undercutting the ball  backspin
Ball100 downward
D = center-to-center offset
Bat 100 upward
250
trajectories
200
“vertical sweet spot”
150
1.5
1.0
2.0
100
0.5
50
What’s going on here??
0
-100
0.75
0.75
0.25
0
100
0
200
300
35
400
36
Another familiar result:
bat tilted downward
bat hits under ball:
popup to opposite field
Catcher’s View
bat hits over ball:
grounder to pull field
37
Putting it all Together:
Can curveball be hit farther
than fastball?
• Bat-Ball Collision Dynamics
–
A fastball will be hit faster
–
A curveball will be hit with more backspin
38
curveball can be hit with more backspin: WHY?
Fastball with backspin
Fastball: spin must reverse
Curveball: spin doesn’t reverse
Curveball with topspin
39
Net effect: backspin larger for curveball
Can Curveball Travel Farther than
Fastball?
• Bat-Ball Collision Dynamics
–
A fastball will be hit faster
–
A curveball will be hit with more backspin
• Aerodynamics
–
A ball hit faster will travel farther
–
Backspin increases distance
• Which effect wins?
• Curveball, by a hair!
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Work in Progress
• Collision experiments & calculations to
elucidate trampoline effect
• New studies of aerodynamics using Doppler
radar
• Experiments on high-speed oblique
collisions
• A book, with Aussi Rod Cross
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Final Summary
• Physics of baseball is a fun application of basic
(and not-so-basic) physics
• Check out my web site if you want to know more
– webusers.npl.uiuc.edu/~a-nathan/pob
– a-nathan@uiuc.edu
• Thanks for your attention and go Red Sox!
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