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Physics and Baseball:Having Your Cake and Eating it Too
Thanks to J. J. Crisco & R. M. GreenwaldMedicine & Science in Sports & Exercise
34(10): 1675-1684; Oct 2002
Alan M. NathanDepartment of Physics
University of Illinois
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How Does a Physicist Analyze the Game of Baseball?
October 27, 2004:the day the curse was
broken
Having Your Cake and Eating it Too
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1927
Solvay Conference:
Greatest physics team
ever assembled
Baseball and Physics
1927 Yankees:
Greatest baseball team
ever assembled
MVP’s
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A great book to read….
“Our goal is not to reform the game but to understand it.
“The physicist’s model of the game must fit the game.”
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Outline
• How does a baseball bat work?
• The flight of a baseball.
• Leaving the no-spin zone.
• Putting it all together.
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“You can observe a lot by watching”
Champaign News-Gazette
CE Composites
--Yogi Berra
Easton Sports
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Description of Ball-Bat Collision• forces large, time short
– >8000 lbs, <1 ms
• ball compresses, stops, expands– KEPEKE– bat recoils
• lots of energy dissipated (“COR”)– distortion of ball – vibrations in bat
• to hit home run….– large batted ball speed
• 100 mph~400 ft, each additional mph ~ 5-6’
– optimum take-off angle (300-350)– lots of backspin
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BBS = q vball + (1+q) vbat
Conclusion:
vbat matters much more than vball
• 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
BBS 0.2 vball + 1.2 vbat
Kinematics of Ball-Bat Collision
vball vbat
BBS
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Kinematics of Ball-Bat Collision
ball bat
e-rq =
1+re-r 1+e
BBS = v v1+r 1+r
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)
q=0.20
BBS = q vball + (1+q) vbatvball vbat
BBS
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Collision Efficiency q Can Be Measured
• Air cannon fires ball onto stationary bat
• q = vout/vin
• Used by NCAA, ASA, … to regulate/limit performance of bats
Sports Sciences Lab @ WSU
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Accounting for COR:
Dynamic Model for Ball-Bat CollisionAMN, Am. J. Phys, 68, 979 (2000)
• Collision excites bending vibrations in bat
– hurts! breaks bats
– dissipates energy • lower COR, BBS
• Dynamic model of collision– Treat bat as nonuniform beam– Treat ball as damped spring
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Modal Analysis of a Baseball Batwww.kettering.edu/~drussell/bats.html
0
0.05
0.1
0.15
0 500 1000 1500 2000 2500
FFT(R)
frequency (Hz)
179
582
1181
1830
2400
frequency
-1.5
-1
-0.5
0
0.5
1
0 5 10 15 20
R
t (ms)
time
0 5 10 15 20 25 30 35
f1 = 179 Hz
f2 = 582 Hz
f3 = 1181 Hz
f4 = 1830 Hz
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Vibrations, COR, and the “Sweet Spot”
Evib
vf
e
+
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0
20
40
60
80
100
120
0 5 10 15
e
vf (mph)
distance from tip (inches)
nodes4 3 2 1
Strike bat here
best performance & feel
@ ~ node 2
•
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• strike bat in barrel—look at response in handle
• 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
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
0 1 2 3 4 5
v (m/s)
t (ms)
Independence of End Conditions
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q independent of end conditions:
experimental proof
0
0.05
0.1
0.15
0.2
0.25
25 26 27 28 29 30 31 32
Collision Efficiency
distance from knob (inches)
"normal" bat
normal + 3 oz in knob
Conclusion: mass added in knob has no effect on collision efficiency (q)
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Vibrations and Broken Bats
movie
0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000
pitcher
catcher
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Aluminum has thin shell – Less mass in barrel
--lower MOI, higher bat speed, easier to control --but less effective at transferring energy --for many bats cancels
» just like corked wood bat
– “Hoop modes” • trampoline effect • “ping”
Does Aluminum Outperform Wood?
demo
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A Closer Look at Hoop Modes
“hoop” modes: cos(2)
“ping”
Thanks to Dan Russell
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•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”—confirmed by experiment
…and higher BBS
• Also seen in golf, tennis, …
The “Trampoline” Effect:A Simple Physical Picture
demo
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0.40
0.45
0.50
0.55
0.60
0.65
0.70
500 1000 1500 2000
COR-modelCOR-expt
COR
fhoop
(Hz)
Softball Data and Model
Conclusion: essential physics understood
Russell, Smith, AMN
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Does Aluminum Outperform Wood?
YES!
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Forces on a Baseball in Flight• Gravity• Drag (“air resistance”)• Lift (or “Magnus”)
v
ω
mg
Fd
FL
Courtesy, Popular Mechanics
Fd=½ CDAv2
-v direction
(ω v) direction FL = ½ CLAv2
direction leading edge is turning
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Typical values of drag and lift
0
0.5
1
1.5
2
0 25 50 75 100 125 150Speed in mph
Drag/Weight
Lift/Weight@1800 rpm
“Drag crisis?”
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Effect of Drag and Lift on Trajectories
• drag effect is huge
• lift effect is smaller but significant
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700
distance (ft)
no drag or lift
drag, no lift drag and lift
v
ω
mg
Fd
FL (Magnus)
250
50
100
150
200
250
300
350
400
10 20 30 40 50 60 70 80 90
Range (ft)
(deg)
Range vs.
2000 rpm
0 rpm
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
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700
distance (ft)
no drag or lift
drag, no lift
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Some Effects of Lift
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700
distance (ft)
no drag or lift
drag, no lift drag and lift
v
ω
mg
Fd
FL (Magnus)
• 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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What’s the Deal with the Gyroball?
Courtesy, The New York TImes
Courtesy, Ryutaro Himeno
Daisuke Matsuzaka:Does he or doesn’t he?
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How Far Did That Home Run Travel?
• Ball leaves bat• Hits stands D from home plate, H above
ground• How far would it have gone if no obstruction?
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0
50
100
150
200
0 100 200 300 400 500horizontal distance (ft)
0
5
10
15
20
25
30
35
40
390 400 410 420 430 440 450 460horizontal distance (ft)
400 ft/30 ftRange=415-455Time can resolve
4 s
5 s7 s
See www.hittrackeronline.com
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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?
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Oblique Collisions:Leaving the No-Spin Zone
Oblique friction spin
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
320
50
100
150
200
250
-100 0 100 200 300 400
1.5
0
0.25
0.5 0.75
1.02.0
0.75
Undercutting the ball backspin
Ball100 downward
Bat 100 upward
D = center-to-center offset
trajectories
“vertical sweet spot”
What’s going on here??
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• Bat-Ball Collision Dynamics– A fastball will be hit faster
– A curveball will be hit with more backspin
Putting it all Together:Can curveball be hit farther
than fastball?
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Net effect: backspin larger for curveball
Fastball: spin must reverse
curveball can be hit with more backspin: WHY?
Fastball with backspin
Curveball: spin doesn’t reverse
Curveball with topspin
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• 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!
Can Curveball Travel Farther than Fastball?
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Work in Progress
• Collision experiments & calculations to elucidate trampoline effect
• New studies of aerodynamics
• Experiments on high-speed oblique collisions
• A book, with Aussi Rod Cross
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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– webusers.npl.uiuc.edu/~a-nathan/pob– [email protected]
• Thanks for your attention and go Red Sox!