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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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the physics of hitting a home run

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


1927 Yankees:

Greatest baseball team

ever assembled


Solvay Conference:

Greatest physics team

ever assembled


Baseball and Physics

UBC Colloquium 10/5/06

adair s book an excellent reference
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

the physics of hitting a home run4
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


“You can observe a lot by watching”

Champaign News-Gazette

--Yogi Berra

Easton Sports


UBC Colloquium 10/5/06


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





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






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

collision efficiency q can be measured
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


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


f1 = 179 Hz

f3 = 1181 Hz

f2 = 582 Hz

f4 = 1830 Hz



Modal Analysis of a Baseball Bat

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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


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


q independent of end conditions:

experimental proof

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

UBC Colloquium 10/5/06

why does aluminum outperform wood
Why Does Aluminum Outperform Wood?
  • Aluminum has thin shell
  • Hoop modes give “trampoline” effect
    • larger COR, vf

UBC Colloquium 10/5/06


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

the trampoline effect a closer look
The Trampoline Effect: A Closer Look

“hoop” modes: cos(2)

Thanks to Dan Russell


UBC Colloquium 10/5/06


Wood vs. Aluminum:

Where Does the Energy Go?

UBC Colloquium 10/5/06

the trampoline effect a closer look19
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


Softball Data and Model

essential physics understood

UBC Colloquium 10/5/06

aerodynamics of a baseball


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

effect of drag and lift on trajectories
Effect of Drag and Lift on Trajectories




  • drag effect is huge
  • lift effect is smaller but significant

UBC Colloquium 10/5/06

some effects of drag
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

some effects of lift




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

new experiment at illinois
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

motion capture experiment joe hopkins lance chong hank kaczmarski amn

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


Typical Motion Capture Datameasure spin, CM trajectory

CM trajectory

Note: topspin  ay > g

UBC Colloquium 10/5/06


Results for Lift Coefficient CL

FL= 1/2ACLv2


100 mph, 2000 rpm


Conclusion: data qualitatively consistent (~20%)

UBC Colloquium 10/5/06

baseball aerodynamics things i would like to know better
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

oblique collisions leaving the no spin zone
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

undercutting the ball backspin

Ball100 downward

D = center-to-center offset

Bat 100 upward

Undercutting the ball  backspin


“vertical sweet spot”

UBC Colloquium 10/5/06

putting it all together can curveball be hit farther than fastball
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


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

can curveball travel farther than fastball
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

work in progress
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

final summary
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