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1
Baseball and Physics:Where Albert Pujols meets Albert Einstein
---Alan Nathan
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Baseball and PhysicsWhere Albert Pujols meets Albert Einstein
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Albert Einstein and Baseball
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Einstein--“Mr. Berg, you teach me baseball and I’ll teach you the theory of relativity.”Then after some thought….“No, we must not. You will learn about relativity faster than I learn baseball.”
Albert Einstein, Moe Berg, and baseball
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A good book to read….
“…the physics of baseball is not the clean, well-defined physics of fundamental matters. Hence conclusions must depend on approximations and estimates. But estimates are part of the physicist’s repertoire...”
“The physicist’s model of the game must fit the game.”
“Baseball is not rocket science. It’s much harder.”
Prof. Bob Adair
relativity
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Topics I Will Cover
• The ball-bat collision– How a bat works– Wood vs. aluminum
• The flight of the baseball– Drag, lift, and all that– New tools for baseball analysis
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“You can observe a lot by watching”---Yogi Berra• forces large, time short
– >8000 lbs, <1 ms
• ball compresses, stops, expands– like a spring: KEPEKE
– bat recoils
• lots of energy dissipated– distortion of ball
– vibrations in bat
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• pitch speed
• bat speed
• “collision efficiency”: a property of the ball and bat
BBS = q vpitch + (1+q) vbat
• typical numbers: q = 0.2 1+q = 1.2
example: 85 + 70 gives 101 mph (~400’)
• vbat matters much more than vpitch!
– Each mph of bat speed worth ~6 ft
– Each mph of pitch speed worth ~1 ft
What Determines Batted Ball Speed?
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Kinematics of Ball-Bat Collision
BBS
eff
eff
e-m/Mq =
1 m/M
1. m/Meff = ball mass/effective bat mass 0.25
bat recoil2. e = elasticity of collision 0.50
energy dissipation
For m/Meff <<1 and e1, q1
BBS = q vpitch + (1+q) vbat
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1. Effective Bat Mass
Meff “Swing Weight”: related to MOI about the handle
Larger less recoil to bat larger q
Larger smaller swing speed
effpitch bat
eff eff
e-m/M 1 eBBS v v
1 m/M 1 m/M
Batters seem to prefer lower MOI bats sacrificing power for “quickness”
Cross and AMN, Sports Technology 2, 7-15 (2009)
40
42
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46
48
50
8.5 9 9.5 10 10.5 11 11.5
I6"
(103 oz-in2)
knob
(rad/s)
y = m1*(9/m0)^m2
ErrorValue
0.392146.218m1
0.0574220.28747m2
NA3.8574Chisq
NA0.93138R
Crisco/Greenwald Batting Cage Study
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Is There an Advantage to “Corking” a Bat?
Based on best experimental data available:…for “harder” hit: no
…for frequency of good contact: probably
Sammy Sosa, June 2003
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2. e = ball-bat coefficient of restitution(bbcor)
• 1 - e2 = fraction of CM energy dissipated– ~75%!
• Joint property of ball and bat– Most of energy loss is in ball
– But the bat matters• Vibrations decrease e• Trampoline effect increase e
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Vibrations and the ball-bat collision
outside “sweet spot”
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Studying the Vibrations of a Baseball Batwww.kettering.edu/~drussell/bats.html
0
0.05
0.1
0.15
0 500 1000 1500 2000 2500
FFT(R)
frequency (Hz)
179
582
1181
1830
2400
frequency
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20
R
t (ms)
time
0 5 10 15 20 25 30 35
f1 = 179 Hz
f2 = 582 Hz
f3 = 1181 Hz
f4 = 1830 Hz
15
20
-2 0
-1 5
-1 0
-5
0
5
10
15
20
0 5 10 15 20 25 30 35
y
z
y
t)F(z, t
yA
z
yEI
z 2
2
2
2
2
2
Dynamics of the Bat-Ball CollisionAMN, AJP 68, 979-990 (2000)
• Solve eigenvalue problem for normal modes • Model ball-bat force F• Expand y in normal modes• Solve coupled equations of motion for ball, bat• Energy budget:
KE of ball (batted ball speed) recoil of bat dissipation in ball vibrations in bat
0
2000
4000
6000
8000
1 104
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
force (pounds)
compression (inches)
approx quadratic
F=kxn
COR
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Vibrations, BBCOR, and the “Sweet Spot”
Evib
vf
e
+at ~ node 2
vibrations minimized
COR maximized
BBS maximized
best “feel”
0.1
0.2
0.3
0.4
0.5
0.6
0
10
20
30
40
50
0 5 10 15
BB
CO
R
vibr
atio
n fr
actio
n
distance from tip (inches)
Ball-Bat COR
vibrational energy
1234nodes
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• strike bat on barrel—look at movement in handle
• handle moves only after ~0.6 ms delay
• collision nearly over by then
• nothing on knob end matters• size, shape, hands, grip• boundary conditions
• confirmed experimentally
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5
v (m/s)
t (ms)
Independence of End Conditions
Batter could drop bat just before contact and it would have no effect on ball!!!
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BBCOR and the Trampoline Effect(hollow bats)
The Ping!
Lowest Hoop (or wineglass) Mode
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• BBCOR increases with … elasticity of ball (~0.5) elasticity of bat (~1)
relative stiffness ~ kball/kbat
• BBCOR(Al)/BBCOR(wood) unregulated, can be very large Little League <1.15 NCAA < 1.0 (!)
The “Trampoline” Effect:A Simple Physical Picture
0.40
0.45
0.50
0.55
0.60
0.65
0.70
500 1000 1500 2000
COR-modelCOR-expt
COR
fhoop
(Hz)
change kbat
wood
alum
Change kball
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Forces on a Spinning Baseball in Flight
D2
D C1
ˆF = - ρAv v2
2LM
1ˆ ˆF = ρAv (ωC v)
2
v
ω
mgFD
FM
• Drag slows ball down
• Magnus + mg deflects ball from straight line
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Real vs. “Physics 101” Trajectory: Effect of Drag and Magnus
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What do we know about CD?(mainly from pitch tracking)
Depends on ….
• v0 (Reynold’s Number)
• surface “roughness”?• seam orientation?• spin?
Dedicated TrackManexperiment@Safeco, Oct. 2008
StL, Sept. 2009PITCHf/xTrackMan
• Good approximation: Cd = 0.35±0.05 in range 60-100 mph
• No steep “drag crisis”• More dedicated experiments in progress
23
What do we know about CL?(mainly from high-speed motion analysis)
Depends on ….• spin parameter S R/v• v @ fixed S?
• best evidence is “no”, in region of 50-100 mph
• seam orientation?
Good approximation: CL S R/v in range 0.05-0.30
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New tools to study flight of baseball
• PITCHf/x and HITf/x– Video tracking
• TrackMan – Doppler radar tracking
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PITCHf/x and HITf/x• Two video cameras @60 fps
– “high home” and “high first”– tracks every pitch in every MLB ballpark
• all data publicly available on web!
– tracks initial trajectory of batted ball
• Used for analysis, TV broadcasts, MLB Gameday, etc.
Image, courtesy of Sportvision
Marv White, Physics,UIUC, 1969Marv White, Physics,UIUC, 1969
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TrackMan• Doppler radar to measure radial velocity
– dr/dt r(t)
• 3-detector array to measure phase– two angles (t), (t)
• Together these give full 3D trajectory• Spin modulates to give sidebands
– spin frequency
TRANSMITTER
RECEIVER 1
FTX
FTX
FRX1fd1
(fd1 ) - (fd2) = 2**sin()*D*FTX/cRECEIVER 2FRX2
fd2
FTX
D
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• Minimal parametrization of the trajectory– Constant acceleration works very well for pitched ball
– Constant “jerk” works for most batted balls
• Determining Magnus acceleration– “spin movement” important for studying pitching
• Keeping everyone honest– Laws of physics cannot be violated– Recognizing errors– Measurements have uncertainties!– Dealing with imperfect data
So what good is a physicist in all this?
0 0(r , v ,a)
0 0 0(r , v ,a , da/dt)
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Baseball Analysis:Using PITCHf/x to discover how
pitchers do what they do
“Hitting is timing. Pitching is upsetting timing.”
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home plate
Ex 1: Mariano Rivera: Why is he so good??
Three Reasons: Location, Location, Location
Home Runs
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Ex 2: “Late Break”: Truth or MythMariano Rivera’s Cut Fastball
View from above:actual trajectory --------linear extrapolation - - - -
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Josh Kalk, THT, 5/22/08
Ex 2a: What makes an effective slider
This slider is very effective since it looks like a fastball for over half the trajectory, then seems to drop at the last minute (“late break”).
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1
2
3
4
5
6
7
0 10 20 30 40 50
C. C. Sabathia: FB vs. Slider
Distance from home plate (ft)
95 mph fastball
82 mph slider
~4 inches
~12 inches
side view
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Ex 3: A Pitcher’s Repertoire
Catcher’s View
4-seam fastball
2-seam fastballchangeup curveball
slider/cutter
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Ex 4 Jon Lester vs. Brandon Webb
Brandon Webb is a “sinkerball” pitcher:Almost no rise on his fastball
15 inches
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Ex 5 The Knuckleball
Tim Wakefield is a knuckleball pitcher:Chaotic Movement
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Learning About Batted Balls
• HITf/x– Initial part of trajectory– All April 2009 data available
• TrackMan– Full trajectory– Limited data from StL, Sept. 2009
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TrackMan Data from StL, 2009
R vs. v0 R vs. 0
USEFUL BENCHMARK400 ft @ 103 mph
~5 ft per mph
peaks @ 25o-35o
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What Constitutes a Well-Hit Ball?
w/o home runs
home runs
HR
BABIP V0>90
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Putting Spin on Batted Balls• in front or behind sidespin
– sideways Magnus force– fly balls break toward foul pole
friction
normal force
0
50
100
150
200
0 50 100 150 200
foul line
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• undercutting/overcutting backspin/topspin
Magnus force is up/down
Topspin makes line drives nose-dive
Backspin keeps fly ball in air longer
Tricky popups to infield
friction
normal forcev
0
50
100
150
200
250
-100 0 100 200 300 400
1.5
0
0.25
0.5 0.75
1.02.0
0.75
???
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Paradoxical PopupsAJP 76, 723-729 (2008)
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Combining HITf/x with Hittracker• HITf/x v0,,
• Hittracker (Greg Rybarczyk, hittrackeronline.com)– Landing point– Flight time
• Together these constrain the full trajectory
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HITf/x+hittracker Analysis: The “carry” of a fly ball
• Motivation: does the ball carry especially well in the new Yankee Stadium? • “carry” ≡ (actual distance)/(vacuum distance)
for same initial conditions
(379,20,5.2)
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HITf/x + hittracker Analysis:4354 HR from 2009
Denver
Cleveland Yankee Stadium
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Work in Progress• Collision experiments & calculations to
elucidate trampoline effect
• New studies of drag and Magnus
• Experiments on high-speed oblique collisions– To quantify spin on batted ball
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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
– go.illinois.edu/physicsofbaseball– [email protected]
• I am living proof that knowing the physics doesn’t help you play the game better!
@ Red Sox Fantasy Camp, Feb. 1-7, 2009
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HITf/x + hittracker Analysis:4354 HR from 2009
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CD: One Final Thought
Correlations suggestive of variations in baseball
PFX TM
PFX-TM
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Extract sidespin vs. from trajectoryCF
RF
break to right
break to left
LF
• Balls break toward foul pole• Break increases with angle• Ball hit to CF slices
LHH/RHH asymmetry Tilt in bat
RF
RHH
LHHLF RF
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Is the Baseball “Juiced”?
Is COR larger than it used to be?
• 1975 and 2004 equal to few %• No evidence for juiced ball
Measurements with high-speed cannon• COR=rebound speed/initial speed• 1975 vs. 2004
50
Example: Pitching at High Altitude
10%
loss of velocity
total movement12”
7.5%
8”
PITCHf/x data contain a wealth of information about drag and lift!
Toronto
Toronto
Denver
Denver