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Physics and Baseball:Having Your Cake and Eating it Too

Thanks to J. J. Crisco & R. M. GreenwaldMedicine & Science in Sports & Exercise

34(10): 1675-1684; Oct 2002

Alan M. NathanDepartment of Physics

University of Illinois

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How Does a Physicist Analyze the Game of Baseball?

October 27, 2004:the day the curse was

broken

Having Your Cake and Eating it Too

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1927

Solvay Conference:

Greatest physics team

ever assembled

Baseball and Physics

1927 Yankees:

Greatest baseball team

ever assembled

MVP’s

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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.”

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Outline

• How does a baseball bat work?

• The flight of a baseball.

• Leaving the no-spin zone.

• Putting it all together.

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“You can observe a lot by watching”

Champaign News-Gazette

CE Composites

--Yogi Berra

Easton Sports

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Description of Ball-Bat Collision• forces large, time short

– >8000 lbs, <1 ms

• ball compresses, stops, expands– KEPEKE– 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

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BBS = q vball + (1+q) vbat

Conclusion:

vbat matters much more than vball

• 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

Kinematics of Ball-Bat Collision

vball vbat

BBS

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Kinematics of Ball-Bat Collision

ball bat

e-rq =

1+re-r 1+e

BBS = v v1+r 1+r

r = mball /Mbat,eff : bat recoil factor = 0.25

(momentum and angular momentum conservation)---heavier is better but…

e: “coefficient of restitution” 0.50 (energy dissipation—mainly in ball, some in bat)

q=0.20

BBS = q vball + (1+q) vbatvball vbat

BBS

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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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Accounting for COR:

Dynamic Model for Ball-Bat CollisionAMN, 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 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

best performance & feel

@ ~ node 2

•

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• strike bat in barrel—look at response in handle

• handle moves only after ~0.6 ms delay

• collision nearly over by then

• 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 4 5

v (m/s)

t (ms)

Independence of End Conditions

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q independent of end conditions:

experimental proof

0

0.05

0.1

0.15

0.2

0.25

25 26 27 28 29 30 31 32

Collision Efficiency

distance from knob (inches)

"normal" bat

normal + 3 oz in knob

Conclusion: mass added in knob has no effect on collision efficiency (q)

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Vibrations and Broken Bats

movie

0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000

pitcher

catcher

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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”

Does Aluminum Outperform Wood?

demo

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A Closer Look at Hoop Modes

“hoop” modes: cos(2)

“ping”

Thanks to Dan Russell

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•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, …

The “Trampoline” Effect:A Simple Physical Picture

demo

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0.40

0.45

0.50

0.55

0.60

0.65

0.70

500 1000 1500 2000

COR-modelCOR-expt

COR

fhoop

(Hz)

Softball Data and Model

Conclusion: essential physics understood

Russell, Smith, AMN

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Does Aluminum Outperform Wood?

YES!

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Forces on a Baseball in Flight• Gravity• Drag (“air resistance”)• Lift (or “Magnus”)

v

ω

mg

Fd

FL

Courtesy, Popular Mechanics

Fd=½ CDAv2

-v direction

(ω v) direction FL = ½ CLAv2

direction leading edge is turning

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Typical values of drag and lift

0

0.5

1

1.5

2

0 25 50 75 100 125 150Speed in mph

Drag/Weight

Lift/Weight@1800 rpm

“Drag crisis?”

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Effect of Drag and Lift on Trajectories

• drag effect is huge

• lift effect is smaller but significant

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift drag and lift

v

ω

mg

Fd

FL (Magnus)

250

50

100

150

200

250

300

350

400

10 20 30 40 50 60 70 80 90

Range (ft)

(deg)

Range vs.

2000 rpm

0 rpm

Some Effects of Drag

• Reduced distance on fly ball

• Reduction of pitched ball speed by ~10%

• Asymmetric trajectory:– Total Distance 1.7 x distance

at apex

• Optimum home run angle ~30o-35o

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift

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Some Effects of Lift

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

distance (ft)

no drag or lift

drag, no lift drag and lift

v

ω

mg

Fd

FL (Magnus)

• 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.

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What’s the Deal with the Gyroball?

Courtesy, The New York TImes

Courtesy, Ryutaro Himeno

Daisuke Matsuzaka:Does he or doesn’t he?

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How Far Did That Home Run Travel?

• Ball leaves bat• Hits stands D from home plate, H above

ground• How far would it have gone if no obstruction?

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0

50

100

150

200

0 100 200 300 400 500horizontal distance (ft)

0

5

10

15

20

25

30

35

40

390 400 410 420 430 440 450 460horizontal distance (ft)

400 ft/30 ftRange=415-455Time can resolve

4 s

5 s7 s

See www.hittrackeronline.com

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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

320

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 going on here??

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• Bat-Ball Collision Dynamics– A fastball will be hit faster

– A curveball will be hit with more backspin

Putting it all Together:Can curveball be hit farther

than fastball?

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Net effect: backspin larger for curveball

Fastball: spin must reverse

curveball can be hit with more backspin: WHY?

Fastball with backspin

Curveball: spin doesn’t reverse

Curveball with topspin

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• 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!

Can Curveball Travel Farther than Fastball?

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Work in Progress

• Collision experiments & calculations to elucidate trampoline effect

• New studies of aerodynamics

• 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!