The Physics of Hitting a Home Run

UW Colloquium 10/31/05 1

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

34(10): 1675-1684; Oct 2002

Alan M. Nathan,University of Illinoiswww.npl.uiuc.edu/~a-nathan/pob

a-nathan @uiuc.edu

The Physics of Hitting a Home Run

http://ipsapp002.lwwonline.com/content/getfile/2320/1359/21/abstract.htm

http://ipsapp002.lwwonline.com/content/getfile/2320/1359/21/abstract.htm


1927 Solvay Conference:

Greatest physics team ever assembled

Baseball and Physics

1927 Yankees:Greatest baseball team

ever assembled

MVP’s


“Hitting is timing; pitching isupsetting timing”

Hitting the Baseball:

the most difficult feat in sports

“Hitting is fifty percent above the shoulders”

1955 Topps cards from my personal collection


Graphic courtesy of Bob Adair and NYT

Hitting and Pitching, Thinking and Guessing


Example: Tim Wakefield’s Knuckleball


1. How does a baseball bat work?

2. Why does aluminum outperform wood?

3. How does spin affect flight of baseball?

4. Can a curveball be hit farther than a

fastball?

The Physics of Hitting a Home Run


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

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

– KEPEKE– bat bends & compresses

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

• to hit home run….– large hit ball speed– optimum take-off angle– lots of backspin

Courtesy of CE Composites


vf = q vball + (1+q) vbat

Conclusion:

vbat matters much more than vball

• q “Collision Efficiency” • 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

vf 0.2 vball + 1.2 vbat

vball vbat

vf

Kinematics of Ball-Bat Collision



f ball bat

e-rq = 1+r

e-r 1+ev = v v1+r 1+r

r = mball /Mbat,eff : bat recoil factor = 0.25(momentum and angular momentum conservation)

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

vball vbat

vf

q=0.20



f ball bat e-r 1+ev = v v1+r 1+r

• r = mball /Mbat,eff: bat recoil factor = 0.25(momentum and angular momentum conservation)

• heavier bat better but…


The Ideal Bat Weight or Iknob

60

70

80

90

100

110

120

20 30 40 50 60

n=0constant v

bat

n=0.5constant bat KE

vbat

= 65 mph x (32/Mbat

)n

Mbat

(oz)

vf (mph)

n=0.31 (expt)

Observation: Batters prefer lighter bats

Experiments:knob ~ (1/Iknob)0.3


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• lower vf


The Details: A Dynamic Model2 2 2

2 2 2

y y A F - EI t x x

:nonuniform beam

-2 0

-1 5

-1 0

-5

0

5

10

15

20

0 5 10 15 20 25 30 35

20

y

z

y

• Step 1: Solve eigenvalue problem for free vibrations

• Step 2: Nonlinear lossy spring for ball-bat interaction F(t)

• Step 3: Expand in normal modes and solve

yA xyEI

x n

2n2

n2

2

2

22n n

n n n n2n

d q F(t) y ( )y( ) q ( )y ( ) qdt A

zx,t t x


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


Some Interesting Insights:Bat Recoil, Vibrations, COR, and “Sweet Spot”

Evib

vf

e

Node of 1nd mode

+

~ 1 ms only lowest 4 modes excited

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

-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)


Experimental Data: Dependence of COR on Impact Location

ball incident on bat at rest

Conclusion: essential physics under control

0.25

0.30

0.35

0.40

0.45

0.50

0.55

23 24 25 26 27 28 29 30 31

e

distance from knob (inches)

flexible bat

rigid bat

Louisville Slugger R161 Wood Batv

i=100 mph


• handle moves only after ~0.6 ms delay

• collision nearly over by then

• nothing on knob end matters• size, shape• boundary conditions• hands

-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


0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000

pitcher

catcher

Vibrations and Broken Bats

outside inside

node


Aluminum has thin shell – Less mass in barrel

–easier to swing and control –but less effective at transferring energy –for many bats cancels

– Hoop modes –trampoline effect –larger COR

Why Does Aluminum Outperform Wood?


•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”

• Also seen in golf, tennis, …

The “Trampoline” Effect:A Simple Physical Picture


The Trampoline Effect: A Closer Look

“hoop” modes: cos(2) • k (t/R)3: hoop mode largest in barrel

• f2 (1-3 kHz) < 1/ 1kHz energy mostly restored

(unlike bending modes)

“ping”

Thanks to Dan Russell


0.40

0.45

0.50

0.55

0.60

0.65

0.70

500 1000 1500 2000

COR-modelCOR-expt

COR

fhoop

(Hz)

Data and Model

to optimize….• kbat small• fhoop > 1

essential physics understood


Effect of Spin on Baseball Trajectory

Drag: Fd = ½ CDAv2

-v direction

“Magnus” or “Lift”: FL = ½ CLAv2

(ω v) direction

v

ω

mg

Fd

FL (Magnus)

CD~ 0.2-0.5CL ~ R/v

(in direction leading edge is turning)


New Experiment at Illinois

• Fire baseball horizontally from pitching machine

• Use motion capture to track ball over ~5m of flight and determine x0,y0,vx,vy,,ay

• Use ay to determine Magnus force as

function of v,


Motion Capture ExperimentJoe Hopkins, Lance Chong, Hank Kaczmarski, AMN

Two-wheel pitching machine

Baseball with reflecting dot

Motion Capture System


Experiment: Sample MoCap Datay

z

topspin ay > g

-3000

-2000

-1000

0

1000

2000

-20

0

20

40

60

80

100

120

140

0.00 0.02 0.04 0.06 0.08 0.10 0.12

z (mm)y (mm)

time (sec)

93.6 mph/3040 rpm/1.83g

Z

y

y = ½ ayt2

work in progress


Some Typical Results

0

0.5

1

1.5

2

0 25 50 75 100 125 150Speed in mph

Drag/Weight

Lift/Weight@1800 rpm

0

20

40

60

80

100

0 50 100 150 200 250 300 350 400x (ft)

2000 rpm backspin

no spin

200

250

300

350

400

450

10 15 20 25 30 35 40 45 50

2000 rpm backspin

no spin

Lift …--increases range--reduces optimum angle


Oblique Collisions:Leaving the No-Spin Zone

Friction … • sliding/rolling vs. gripping• transverse velocity reduced, spin increased

vT′ ~ 5/7 vT ~ vT

′/R

Familiar Results• Balls hit to left/right break toward foul line• Topspin gives tricky bounces in infield• Pop fouls behind the plate curve back toward field• Backspin keeps fly ball in air longer

f


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

Undercutting the ball backspinBall100 downward

Bat 100 upward

D = center-to-center offset

trajectories


larger for curveball

-1000

0

1000

2000

3000

4000

5000

6000

0 0.2 0.4 0.6 0.8 1A

2000 rpm topspin

2000 rpm backspin

D (in)

(rpm)

Fastball: spin reverses

Curveball: spin doesn’t reverse


• 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?


Work in Progress

• Collision experiments & calculations to elucidate trampoline effect

• New measurements of lift and drag• Experiments on oblique collisions

– Rod Cross & AMN: rolling almost works at low speed

– AMN: studies in progress at high speed


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– www.npl.uiuc.edu/~a-nathan/pob– [email protected]

• Go Red Sox!

The Physics of Hitting a Home Run

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