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Baseball & Physics:An Intersection of Passions

Alan M. Nathan Department of Physics

University of Illinoisa-nathan@illinois.edu

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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.”

“Our goal is not to reform the game but to understand it.”

“The physics of baseball is not rocket science. It’s much harder”

My friend and mentor, Prof. Bob Adair

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And check out my web site…webusers.npl.uiuc.edu/~pob/a-nathan

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1927 Solvay Conference:

Greatest physics team ever assembled

The Baseball/Physics Connection

1927 Yankees:Greatest baseball team

ever assembled

MVP’s

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Topics I Will Cover

• The ball-bat collision– How a bat works– Wood vs. aluminum– Putting spin on the ball

• The flight of the baseball– Drag, lift, and all that– New tools– How much does a ball “carry”?

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“You can observe a lot by watching”

UMass/Lowell

--Yogi Berra

Easton Sports

Daily Illini

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When ash meets cowhide….• forces large, time short

– >8000 lbs, <1 ms• ball compresses, stops, expands

– like a spring: KEPEKE– bat recoils

• lots of energy dissipated (“COR”)– distortion of ball – vibrations in bat

• to hit home run….– large batted ball speed

• 105 mph~400 ft, each additional mph ~ 5-6’

– optimum take-off angle (300-350)– lots of backspin

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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 = 1.2example: 90 + 70 gives 102 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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What does q depend on?

1. “Swing Weight”: MOI about the handle

– Larger MOI:

• less recoil to bat larger q

• smaller swing speed (usually)

– What is ideal swing weight?

• effect of swing weight on q is easy

• effect of swing weight on vbat is harder

BBS = q vpitch + (1+q) vbat

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The Ideal Bat Weight (or MOI)

Batters seem to prefer lower MOI bats, sacrificing power for “quickness”

BBS (mph)

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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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What does q depend on?

2. Bounciness of ball– “coefficient of

restitution” or COR

– COR= vf/vi

– ~0.5 for baseball• 75% of energy dissipated

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What does q depend on?3. Impact location on bat

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

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Vibrations, COR, and the “Sweet Spot”

Evib

vf

e

+

0.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0

20

40

60

80

100

120

0 5 10 15

e

vf (mph)

distance from tip (inches)

nodes4 3 2 1

Strike bat here

at ~ node 2vibrations minimizedCOR maximizedBBS maximizedbest “feel”

Note: COP is irrelevant to feel and performance

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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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Aluminum has thin shell – Less mass in barrel—lower MOI

--higher bat speed and quickness --but smaller q --for many bats nearly cancels

» just like corked wood bat– “Hoop modes”

• trampoline effect • “ping”

Does Aluminum Outperform Wood?

YES!

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• Two springs mutually compress• Energy shared between “ball spring” and “bat spring”

Sharing depends on relative “stiffnesses” of springs

• Energy stored in ball mostly dissipated (~80%!)• Energy stored 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, …

The “Trampoline” Effect:A Simple Physical Picture

demo

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Flight of the Baseball• Gravity• Drag (“air resistance”)• Lift (or “Magnus”)

v

ω

mg

Fd

FM

Courtesy, Popular Mechanics

Fd=½ CDAv2

-v direction

FM = ½ CLAv2 CL = CMR/v

(ω v) direction direction leading edge is turning

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Real vs. “Physics 101” Trajectory: Effect of Drag

• Reduced distance on fly ball

• Reduction of pitched ball speed by 8-10 mph

• Asymmetric trajectory:– Total Distance 1.7 x

distance at apex

• Optimum home run angle ~30o

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700distance (ft)

no drag or lift

drag, no lift

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Some Effects of Spin

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700distance (ft)

no drag or lift

drag, no lift drag and lift

• 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

mg

Fdrag

FMagnus

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Oblique Collisions and Spin

Movie clip

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friction

normal forcev

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Some Familiar Effects• Balls hit to left/right break toward foul line• Topspin makes line drives nose-dive• Backspin keeps fly ball in air longer• Tricky popups to infield

View from above

friction

velocity

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Another familiar result:

Balls hit to CF slice

Spin axis

From LHH, ball will curve toward LF

270

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

Undercutting the ball backspin

Ball100 downward

Bat 100 upward

D = center-to-center offset

trajectories

“vertical sweet spot”

What’s this all about?

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Paradoxical Popups

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What are we learning from the PITCHf/x system?

A report from the summit

• What is PITCHf/x and how does it work?• What are we learning from it?• Outlook for future

webusers.npl.uiuc.edu/~a-nathan/pob/pitchtracker.htmlsportvision.com/events/pfx.html

This section prepared with help from John Walsh, Mike Fast, Josh Kalk, Dan Brooks, and the good folks at Sportvison, mainly Marv White.

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PITCHf/x is a pitch-tracking system installed in every MLB venue—a joint venture of Sportvision & MLBAM

MLB Gameday

ESPN K-Zone

Fox Trak

MLB Gameday Screen

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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”

• Software to identify and track pitch frame-by- frame in real time full trajectory

lots of other stuff

Image, courtesy of Sportvision

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What kind of “stuff” can one learn?• Pitch speed to ~0.5 mph

– at release and at home plate (they are different!)• Pitch location to ~0.5 inches

– at release and at home plate• “movement” to ~2.0 inches

– both magnitude and direction• Initial velocity direction• Type of pitch

– more on this later • And all these data are freely available online!

3360

65

70

75

80

85

90

70 75 80 85 90 95 100v

0 (mph)

y = m1 * M0ErrorValue

0.000868580.89296m1 NA56.865ChisqNA0.98879R

• Pitched ball loses about 10% of speed between pitcher and batter• Average speed <v> is ~95% of release speed

Example: Pitch Speed--PITCHf/x vs. the gun

PITCHf/x is almost surely more accurate than the gun

v0

vf

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Example: Pitching at High Altitude:Higher <v>, less movement in Denver vs. Toronto

10%

loss of velocity

total movement12”

7.5%

8”

PITCHf/x data contain a wealth of information about drag and lift!

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Example--20k pitches from Anaheim, 2007:Cd vs. v0 <Cd> vs. v0 in 2 mph bins

Drag Coefficient from Pitchf/x

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Drag Coefficient:no evidence for “drag crisis”

0.00

0.10

0.20

0.30

0.40

0.50

0.60

60 65 70 75 80 85 90 95 100

Cd

v (mph)

wind tunnel

Adair

older pitch tracking

Pitchf/x Anaheim 2007

Briggs

Good approximation: Cd = 0.35±0.05 in range 70-100 mph

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>90 mph

80-90 mph

<80 mph

4-seam fastball

2-seam fastball

cutter/slider

curveball

Pitch Classification: LHP Jon Lester, Aug. 3, 2007

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Josh Kalk, THT, 5/22/08

What makes an effective slider?—C. C. Sabathia

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”).

0

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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PITCHf/x tackles the knuckleball – John Walsh

• Classify pitches using vertical and horizontal break plus speed

• Compare “normal” pitcher (C.C. Sabathia) with k-baller (Tim Wakefield)

• “Randomness” of k-ball break is evident in PITCHf/x data

• Example analysis: What happens when knuckleball does not “knuckle”?

• Split k-balls into 3 groups – small, medium, large break

fastball

curve

slider

change

knuckler

Amount of Break

Pitches put in play

OPS against

Small 47 .979Medium

71 .873

Large 79 .684

http://www.hardballtimes.com

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New Tools to Study Trajectories of Batted Balls

• Hitf/x– Uses Pitchf/x cameras to track initial trajectory

• v0,,• Hittracker (www.hittrackeronline.com)

– Measure landing point and flight time for home runs

• Trackman– Phased array Doppler radar– Measure full trajectory and spin

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Example of Hitf/x Analysis:Batting Average for Balls in Play

<-10 -5:0

5:10

15:2

0

25:3

0

35:4

0

>45

theta (deg)

BABIP0.800-1.0000.600-0.8000.400-0.6000.200-0.4000.000-0.200

w/o home runs home runs only

V0

V0

V0

<-10 -5:0

5:10

15:2

0

25:3

0

35:4

0

>45

theta (deg)

BABIP0.800-1.0000.600-0.8000.400-0.6000.200-0.4000.000-0.200

V0

<-10 -5:0

5:10

15:2

0

25:3

0

35:4

0

>45

theta (deg)

HR/balls in play

0.600-0.8000.400-0.6000.200-0.4000.000-0.200

V0

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Example of Hitf/x Analysis:Batting Average for Balls in Play

BABIP:V0>90 mph

w/o home runs

home runs

Goal: Establish “outcome-independent” hitting metrics

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Tracking Everything

Movie clip

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Combining HITf/x with Hittracker

0

100

200

300

400

0 100 200 300 400Horizontal Distance (ft)

(379,20,5.2)

hitf/xhittracker

Full trajectory is constrained by • initial velocity vector• landing point• flight time

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Final example:The “carry” of a fly ball

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The “carry” of a fly ball

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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 & Magnus on seam orientation, surface roughness, …

• What is the time constant for spin decay?

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Work in Progress• Collision experiments & calculations to

elucidate trampoline effect• New studies of aerodynamics• Experiments on high-speed oblique

collisions– To quantify spin on batted ball

• 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@illinois.edu

• I am living proof that knowing the physics doesn’t help you play the game better!

@ Red Sox Fantasy Camp, Feb. 1-7, 2009