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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Adairs Book An Excellent Reference
Our goal is not to reform the game but to
understand it. The physicists model of the
game must fit the game.
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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

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You can observe a lot by watching
--Yogi Berra
Easton Sports
CE Composites
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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 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

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Kinematics of Ball-Bat Collision
vf q vball (1q) vbat

• q ? Collision Efficiency
• Joint 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) ---heavier is better but e
coefficient of restitution ? 0.50
(energy dissipationmainly in ball,
some in bat)
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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 _at_ WSU
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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

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Modal Analysis of a Baseball Bat www.kettering.edu
/drussell/bats.html
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Vibrations, COR, and the Sweet Spot
Strike bat here
Measure response here
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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

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q independent of end conditions experimental
proof
Conclusion mass added in knob has no effect on
collision efficiency (q)
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Why Does Aluminum Outperform Wood?

• Aluminum has thin shell
• Hoop modes give trampoline effect
• larger COR, vf

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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?)

Thanks to Dan Russell
ping
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Wood vs. Aluminum Where Does the Energy Go?
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The Trampoline Effect A Closer Look
to optimize. kbat//kball small and fhoop? gt 1
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Softball Data and Model
essential physics understood
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Aerodynamics of a Baseball
(in direction leading edge is turning)
CD 0.2-0.5 CL R?/v
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Effect of Drag and Lift on Trajectories

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

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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

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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.

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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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Typical Motion Capture Datameasure spin, CM
trajectory
CM trajectory
Note topspin ? ay gt g
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Results for Lift Coefficient CL
FL 1/2?ACLv2 Sr?/v 100 mph, 2000
rpm ?S0.17
Conclusion data qualitatively consistent (20)

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Baseball AerodynamicsThings I would like to
know better

• Better data on drag
• drag crisis?
• Spin-dependent drag?
• Drag for vgt100 mph
• Dependence of drag/lift on seam orientation
• Is the spin constant?

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Oblique CollisionsLeaving 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

demo
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Undercutting the ball ? backspin
trajectories
vertical sweet spot
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Putting it all TogetherCan curveball be hit
farther than fastball?

• Bat-Ball Collision Dynamics
• A fastball will be hit faster
• A curveball will be hit with more backspin

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curveball can be hit with more backspin WHY?
Fastball spin must reverse
Net effect backspin 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 studies of aerodynamics
• Experiments on oblique collisions
• No data!

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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!