1

Physics and Baseball:A Report to Red Sox Nation

Alan Nathan, University of Illinois

2

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

3

1927

Solvay Conference:

Greatest physics team

ever assembled

The Baseball/Physics Connection

1927 Yankees:

Greatest baseball team

ever assembled

MVP’s

4

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

5

“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

6

• 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?

7

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

8

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

44

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

9

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

10

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

11

Vibrations and the ball-bat collision

outside “sweet spot”

12

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

13

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

14

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

15

• 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!!!

16

BBCOR and the Trampoline Effect(hollow bats)

The Ping!

Lowest Hoop (or wineglass) Mode

17

• 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

18

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

19

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

20

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

21

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

22

New tools to study flight of baseball

• PITCHf/x and HITf/x– Video tracking

• TrackMan – Doppler radar tracking

23

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

24

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

25

• Minimal parametrization of the trajectory– Constant acceleration works very well for pitched ball

– Constant “jerk” works for most batted balls

• Keeping everyone honest– Laws of physics cannot be violated– Recognizing and dealing with imperfect data– Measurements have uncertainties!

So what good is a physicist in all this?

0 09 parameters: (r , v ,a)

0 0 012 parameters: (r , v ,a , da/dt)

26

Baseball Analysis:Using PITCHf/x to discover how

pitchers do what they do

“Hitting is timing. Pitching is upsetting timing.”

27

home plate

Ex 1: Mariano Rivera: Why is he so good??

Three Reasons: Location, Location, Location

Home Runs

28

Ex 2: “Late Break”: Truth or MythMariano Rivera’s Cut Fastball

View from above:actual trajectory --------linear extrapolation - - - -

29

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”).

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

30

Ex 3: A Pitcher’s Repertoire

Catcher’s View

4-seam fastball

2-seam fastballchangeup curveball

slider/cutter

31

Ex 4 Jon Lester vs. Brandon Webb

Brandon Webb is a “sinkerball” pitcher:Almost no rise on his fastball

15 inches

32

Ex 5 The Knuckleball

Tim Wakefield is a knuckleball pitcher:Chaotic Movement

33

"Every time that I come here to San Diego, it's always good. Everything moves different. The breaking ball is really nasty, and my fastball moves a lot. So I love it here."

Ex 5 Ubaldo Jimenez Pitching at High Altitude

Denver

vf/v0

DenverDenverDenver

San Diego

San Diego

Denver

Denver

vf/v0

34

Learning About Batted Balls

• HITf/x– Initial part of trajectory– All April 2009 data available

• TrackMan– Full trajectory– Limited data from StL, Sept. 2009

35

TrackMan Data from StL, 2009

R vs. v0 R vs. 0

USEFUL BENCHMARK400 ft @ 103 mph

~5 ft per mph

peaks @ 25o-35o

36

What Constitutes a Well-Hit Ball?

w/o home runs

home runs

HR

BABIP V0>90

37

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

38

• 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

???

39

Paradoxical PopupsAJP 76, 723-729 (2008)

40

Combining HITf/x with Hittracker• HITf/x v0,,

• Hittracker (Greg Rybarczyk, hittrackeronline.com)– Landing point– Flight time

• Together these constrain the full trajectory

41

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300 350distance (ft)

42

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)

43

HITf/x + hittracker Analysis:4354 HR from 2009

Denver

Cleveland Yankee Stadium

44

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

45

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– 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

46

HITf/x + hittracker Analysis:4354 HR from 2009

47

CD: One Final Thought

Correlations suggestive of variations in baseball

PFX TM

PFX-TM

48

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

49

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