Alan M. Nathan,University of Illinois

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The Physics of Hitting a Home Run

• Alan M. Nathan,University of Illinois
• www.npl.uiuc.edu/a-nathan/pob
• a-nathan _at_uiuc.edu

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Baseball and Physics
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Philosophical Remarks (Courtesy of Bob Adair)

• 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
  physicists repertoire...
• The physicists model of the game must fit the
  game.
• Our goal is not to reform the game but to
  understand it.

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Hitting the Baseball the most difficult feat in
sports
1955 Topps cards from my personal collection
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Hitting and Pitching, Thinking and Guessing
Graphic courtesy of Bob Adair and NYT
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Example Tim Wakefields Knuckleball
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The Physics of Hitting a Home Run

• How does a baseball bat work?
• Why does aluminum outperform wood?
• How does spin affect flight of baseball?
• Can a curveball be hit farther than a fastball?

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Brief Description of Ball-Bat Collision

• forces large, time short
• gt8000 lbs, lt1 ms
• ball compresses, stops, expands
• KE?PE?KE
• 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
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Kinematics of Ball-Bat Collision
vf q vball (1q) vbat

• q ? Collision Efficiency
• property of ball bat
• independent of reference frame
• independent of end conditionsmore later
• weakly dependent on vrel
• Superball-wall q ? 1
• Ball-Bat near sweet spot q ? 0.2
• ? vf ? 0.2 vball 1.2 vbat

Conclusion vbat matters much more than vball
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Kinematics of Ball-Bat Collision
r mball /Mbat,eff bat recoil factor ?
0.25 (momentum and angular momentum
conservation) e coefficient of restitution ?
0.50 (energy
dissipationmainly in ball, some in bat)
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Kinematics of Ball-Bat Collision

• r mball /Mbat,eff bat recoil factor ? 0.25
• (momentum and angular momentum conservation)
• heavier bat better but

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The Ideal Bat Weight or Iknob
Experiments ?knob (1/Iknob)0.3
Observation Batters prefer lighter bats
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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
• lower vf

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The Details A Dynamic Model

• 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

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Modal Analysis of a Baseball Bat www.kettering.edu
/drussell/bats.html
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Some Interesting InsightsBat Recoil,
Vibrations, COR, and Sweet Spot

e
vf
Evib
? 1 ms ? only lowest 4 modes excited
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Experimental Data Dependence of COR on Impact
Location ball incident on bat at rest
Conclusion essential physics under control
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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

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Vibrations and Broken Bats
inside
outside
node
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Why Does Aluminum Outperform Wood?

• 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 ??

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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,

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The Trampoline Effect A Closer Look

• hoop modes cos(2?)

• k ? (t/R)3 hoop mode largest
• in barrel
• f2 (1-3 kHz) lt 1/ ? ? 1kHz
• ? energy mostly restored
• (unlike bending modes)

Thanks to Dan Russell
ping
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Data and Model

• to optimize.
• kbat small
• fhoop? gt 1

essential physics understood
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Effect of Spin on Baseball Trajectory
(in direction leading edge is turning)
CD 0.2-0.5 CL R?/v
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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 of
  v, ?

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Motion Capture ExperimentJoe Hopkins, Lance
Chong, Hank Kaczmarski, AMN
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Experiment Sample MoCap Data
topspin ? ay gt g
y ½ ayt2
work in progress
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Some Typical Results
Lift --increases range --reduces optimum angle
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Oblique CollisionsLeaving 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
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Undercutting the ball ? backspin
trajectories
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? ? larger for curveball
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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!

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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

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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
• www.npl.uiuc.edu/a-nathan/pob
• a-nathan_at_uiuc.edu
• Go Red Sox!