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Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002. The Physics of Hitting a Home Run. Alan M. Nathan,University of Illinois a-nathan 1927 Yankees: Greatest baseball team ever assembled.

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Thanks to J. J. Crisco & R. M. Greenwald

Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002

The Physics of Hitting a Home Run

Alan M. Nathan,University of Illinois


UBC Colloquium 10/5/06

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1927 Yankees:

Greatest baseball team

ever assembled


Solvay Conference:

Greatest physics team

ever assembled


Baseball and Physics

UBC Colloquium 10/5/06

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Adair’s Book: An Excellent Reference

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

“The physicist’s model of the game must fit the game.”

UBC Colloquium 10/5/06

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The Physics of Hitting a Home Run

  • How does a baseball bat work?

  • Aerodynamics: flight of a baseball

  • Leaving the no-spin zone

  • Putting it all together

UBC Colloquium 10/5/06

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“You can observe a lot by watching”

Champaign News-Gazette

--Yogi Berra

Easton Sports


UBC Colloquium 10/5/06

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Brief Description of Ball-Bat Collision

  • forces large, time short

    • >8000 lbs, <1 ms

  • ball compresses, stops, expands

    • KEPEKE

    • bat recoils

  • lots of energy dissipated (“COR”)

    • distortion of ball

    • vibrations in bat

  • to hit home run….

    • large hit ball speed (100 mph~400 ft)

    • optimum take-off angle (300-350)

    • lots of backspin

UBC Colloquium 10/5/06

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Kinematics of Ball-Bat Collision

vf = q vball + (1+q) vbat

  • 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

    •  vf 0.2 vball + 1.2 vbat


vbat matters much more than vball

UBC Colloquium 10/5/06

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Kinematics of Ball-Bat Collision

  • 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)

UBC Colloquium 10/5/06

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Collision Efficiency q Can Be Measured

  • Air cannon to fire ball onto stationary bat

  • q = vout/vin

  • Used by NCAA, ASA, … to regulate/limit performance of bats

Sports Sciences Lab @ WSU

UBC Colloquium 10/5/06

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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, vf

  • Dynamic model of collision

    • Treat bat as nonuniform beam

    • Treat ball as damped spring

UBC Colloquium 10/5/06

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f1 = 179 Hz

f3 = 1181 Hz

f2 = 582 Hz

f4 = 1830 Hz



Modal Analysis of a Baseball Bat

UBC Colloquium 10/5/06

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Vibrations, COR, and the “Sweet Spot”

Node of 1nd mode





Strike bat here

Measure response here

UBC Colloquium 10/5/06

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Independence of End Conditions

  • 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

  • UBC Colloquium 10/5/06

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    q independent of end conditions:

    experimental proof

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

    UBC Colloquium 10/5/06

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    Why Does Aluminum Outperform Wood?

    • Aluminum has thin shell

    • Hoop modes give “trampoline” effect

      • larger COR, vf

    UBC Colloquium 10/5/06

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    The “Trampoline” Effect:

    A Simple Physical Picture

    • 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, …

    UBC Colloquium 10/5/06

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    The Trampoline Effect: A Closer Look

    “hoop” modes: cos(2)

    Thanks to Dan Russell


    UBC Colloquium 10/5/06

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    Wood vs. Aluminum:

    Where Does the Energy Go?

    UBC Colloquium 10/5/06

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    The Trampoline Effect: A Closer Look

    Bending Modes vs. Shell Modes

    • k  R4: large in barrel

    •  little energy stored

    • f (170 Hz, etc) > 1/

    •  energy goes into

    • vibrations

    • k  (t/R)3: small in barrel

    •  more energy stored

    • f (2-3 kHz) < 1/ 

    •  energy mostly restored

    to optimize….

    kbat//kball small and fhoop > 1

    UBC Colloquium 10/5/06

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    Softball Data and Model

    essential physics understood

    UBC Colloquium 10/5/06

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    Drag: Fd = ½ CDAv2

    “Magnus” or “Lift”: FL= ½ CLAv2



    Aerodynamics of a Baseball

    (in direction leading edge is turning)

    CD~ 0.2-0.5

    CL ~ R/v

    UBC Colloquium 10/5/06

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    Effect of Drag and Lift on Trajectories




    • drag effect is huge

    • lift effect is smaller but significant

    UBC Colloquium 10/5/06

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    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

    UBC Colloquium 10/5/06

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    Some Effects of 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.

    UBC Colloquium 10/5/06

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    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 ofv, 

    UBC Colloquium 10/5/06

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    Motion Capture System

    Two-wheel pitching machine

    Baseball with reflecting dot

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

    UBC Colloquium 10/5/06

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    Typical Motion Capture Datameasure spin, CM trajectory

    CM trajectory

    Note: topspin  ay > g

    UBC Colloquium 10/5/06

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    Results for Lift Coefficient CL

    FL= 1/2ACLv2


    100 mph, 2000 rpm


    Conclusion: data qualitatively consistent (~20%)

    UBC Colloquium 10/5/06

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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?

    UBC Colloquium 10/5/06

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    Oblique Collisions:Leaving the No-Spin Zone

    Oblique  friction  spin

    transverse velocity reduced

    spin increased

    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


    UBC Colloquium 10/5/06

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    Ball100 downward

    D = center-to-center offset

    Bat 100 upward

    Undercutting the ball  backspin


    “vertical sweet spot”

    UBC Colloquium 10/5/06

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    Putting it all Together:Can curveball be hit farther than fastball?

    • Bat-Ball Collision Dynamics

      • A fastball will be hit faster

      • A curveball will be hit with more backspin

    UBC Colloquium 10/5/06

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    Fastball with backspin

    Curveball: spin doesn’t reverse

    Curveball with topspin

    curveball can be hit with more backspin: WHY?

    Fastball: spin must reverse

    Net effect: backspin larger for curveball

    UBC Colloquium 10/5/06

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    Can Curveball Travel Farther than Fastball?

    • 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!

    UBC Colloquium 10/5/06

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    Work in Progress

    • Collision experiments & calculations to elucidate trampoline effect

    • New studies of aerodynamics

    • Experiments on oblique collisions

      • No data!

    UBC Colloquium 10/5/06

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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 Red Sox!

    UBC Colloquium 10/5/06