1. The Physics of Hitting a Home Run
Alan M. Nathan,University of Illinois
Thanks to J. J. Crisco & R. M. Greenwald
Medicine & Science in Sports & Exercise 34(10): ; Oct 2002
Computer rendition of the swinging bat, the incoming and outgoing ball, and the collision between them, from a study in which high-speed video was used to track the location of the ball and several points on the bat as the batter swung at a ball delivered by a pitching machine. The separation between adjacent frames is 4 ms. Data from images such as these can be used to determine the relationship between the outgoing speed, incoming speed, and bat speed, as a function of impact location along the bat. For example, the above image clearly shows that the outgoing ball, shown as the upper set of images, has a higher speed than the incoming ball, shown as the lower set of images. The purple dot shows the evolution of the instantaneous rotation axis of the bat. Interestingly, just prior to the collision, the rotation axis approximately coincides with the wrist of the lower hand.
UW Colloquium 10/31/05 1

2. Greatest baseball team
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
MVP’s
1927
Solvay Conference:
Greatest physics team
ever assembled
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3. the most difficult feat in sports
Hitting the Baseball:
the most difficult feat in sports
“Hitting is fifty percent above the shoulders”
“Hitting is timing; pitching is
upsetting timing”
1955 Topps cards from my personal collection
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4. Hitting and Pitching, Thinking and Guessing
Greg Maddox: every pitch looks alike for the first 20 ft or so.
Graphic courtesy of Bob Adair and NYT
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5. Example: Tim Wakefield’s Knuckleball
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6. 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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7. Brief Description of Ball-Bat Collision
forces large, time short
>8000 lbs, <1 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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8. Kinematics of Ball-Bat Collision
vball
vbat vf
vf = q vball + (1+q) vbat
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 vbat
Primary dependence of vf on ball, bat speeds is in the relationship…weak dependence of ea itself.
Conclusion:
vbat matters much more than vball
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9. Kinematics of Ball-Bat Collision
vball
vbat vf
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)
q=0.20
Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e,
which depends on rel ball-bat speed.
Mass is effective mass due to extended nature of bat.
Inefficient collision (1 for superball on rigid wall)
Bat speed matters much more than ball speed
UW Colloquium 10/31/05

10. Kinematics of Ball-Bat Collision
r = mball /Mbat,eff: bat recoil factor =  0.25
(momentum and angular momentum conservation)
heavier bat better but…
Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e,
which depends on rel ball-bat speed.
Mass is effective mass due to extended nature of bat.
Inefficient collision (1 for superball on rigid wall)
Bat speed matters much more than ball speed
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11. The Ideal Bat Weight or Iknob
Experiments:
knob ~ (1/Iknob)0.3
NB: Data suggest Iknob matters much more than weight per se.
Observation: Batters prefer lighter bats
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12. Dynamic Model for Ball-Bat Collision
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
So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect the collision process, time slice by time slice, to see what is really going on during the time the ball and bat are in contact. In doing so, we want to try to do a strict accounting for where the energy goes in the collision. So, we want to go beyond kinematics and talk about dynamics. We know that a purely rigid body treatment cannot be right…for example, we know that the collision can excite vibrations in the bat.
The swing:
Early—translation
Later—rotation
The hit
Bat slows down, vibrates
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13. The Details: A Dynamic Model20 y z
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
Cover only briefly
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14. Modal Analysis of a Baseball Bat
f1 = 179 Hz
f2 = 582 Hz
f3 = 1181 Hz
f4 = 1830 Hz
time
frequency
Note: nodes stack up at barrel end-->”sweet spot zone”
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15. Some Interesting Insights: Bat Recoil, Vibrations, COR, and “Sweet Spot”
Node of 1nd mode + e vf
Evib
Impact at various places on barrel, motion at handle near middle of top hand.
Players say that is where the sting occurs. Bottom hand not as sensitive because of damping due to top hand. Happens to nearly coincide with node of lowest mode.
COR vs z
feel
1. Vibrations
No vibrations of 1st mode seen at handle node
Impact at node of 2nd mode: only high-freq. vibration.
At other locations, 560 Hz mode there. That is probably the buzz.
Hands not sensitive to higher frequencies.
2. Maximum of e, max. of vf = minimum of vibrational energy at 2nd node
~ 1 ms  only lowest 4 modes excited
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16. Experimental Data: Dependence of COR on Impact Location
ball incident on bat at rest
For rigid, peak is at CM
For full calculation, peak determined by interplay between rotational recoil and vibrational nodes.
At lowest node, rigid-full
Conclusion: essential physics under control
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17. 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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18. Vibrations and Broken Bats
inside
outside
pitcher
catcher
node
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19. 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 
This summarizes the important properties of bats
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20. 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, …
Ask question:
Which give more “power”: tighter strings or looser strings on tennis racket?
Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline?
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21. 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)
Thanks to Dan Russell
The f-tau relationship is important…the reason why there is no trampoline effect from bending modes--see next slide
Tricks: double wall bats; composite bats
“ping”
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22. essential physics understood
Data and Model
to optimize….
kbat small
fhoop > 1
essential physics understood
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23. Effect of Spin on Baseball Trajectorymg Fd
FL (Magnus) 
Drag: Fd = ½ CDAv2
“Magnus” or “Lift”: FL = ½ CLAv2
(in direction leading edge is turning)
CD~
CL ~ R/v
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24. 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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25. Motion Capture Experiment Joe Hopkins, Lance Chong, Hank Kaczmarski, AMN
Motion Capture System
Two-wheel pitching machine
Baseball with reflecting dot
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26. Experiment: Sample MoCap Datay z
topspin  ay > g
y = ½ ayt2
work in progress
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27. --reduces optimum angle
Some Typical Results
Lift …
--increases range
--reduces optimum angle
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28. 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
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29. Undercutting the ball  backspin
Ball100 downward
Bat 100 upward
D = center-to-center offset
trajectories
Undercut too much: popup with lots of backwpin
Undercut too little: low trajectory with little backspin
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30. Fastball: spin reverses
Curveball: spin doesn’t reverse
  larger for curveball
Undercut too much: popup with lots of backwpin
Undercut too little: low trajectory with little backspin
More friction does not help
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31. 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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32. 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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33. 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!
UW Colloquium 10/31/05