Transcript Juiced Baseballs, Corked Bats, and other Myths of Baseball
Baseball and Mathematics:
It’s More Than Batting Averages ---Alan Nathan 1
The Baseball/Physics Connection
1927 Yankees: Greatest baseball team ever assembled MVP’s 1927 Solvay Conference : Greatest physics team ever assembled 2
A good book to read….
Prof. Bob Adair 3
Another very useful reference…
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Topics I Will Cover
• Dynamics of the ball-bat collision – How a bat works – Wood vs. aluminum • The flight of the baseball – Drag, lift, and all that – Fancy techniques for baseball analysis 5
“You can observe a lot by watching” ---Yogi Berra
• forces large, time short – >8000 lbs, <1 ms • ball compresses, stops, expands – like a spring: KE PE KE – bat recoils • lots of energy dissipated – distortion of ball – vibrations in bat 6
What Determines B atted B all S peed?
• pitch speed • bat speed • “collision efficiency”: a property of the ball and bat BBS = q v pitch + ( 1+q ) v bat • typical numbers: q = 0.2 1+q = 1.2
example: 85 + 70 gives 101 mph (~400’) • v bat matters much more than v pitch !
– Each mph of bat speed worth ~6 ft – Each mph of pitch speed worth ~1 ft 7
The simple stuff: kinematics BBS = q vpitch + ( 1+q ) vbat q = e-m/M eff eff 0.2
1. m/M eff
•
= ball mass/effective bat mass bat recoil
2. e = elasticity of collision
0.50
• •
0.25 “ball-bat coefficient of restitution” (BBCOR)
harder stuff: dynamics!
3. For m/M eff <<1 and e
1, q
1
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Dynamics of Ball-Bat Collision: A Toy Model q ball Mass= 1 2 e A 0.7
0.6
0.5
<< 1 m on M a (1 on 2) 0.4
flexible 0.3
0 2 4 6 4 >> 1 m on M a +M b (1 on 6) 8 bat 10 rigid 9
The Ball-Bat Force (from real data) Force (lb) 10000 8000 6000 4000 2000 0 0 160 mph 80 mph 2 (ms) 1.6
1.2
collision time versus impact speed 0.8
0.4
0 20 40 60 80 100 120 140 impact speed (mph) 0.2
0.4
time (ms) 0.6
0.8
Vibrational modes with f<2 kHz are important
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Experimental Studies of Bat Vibrations www.kettering.edu/~drussell/bats.html
f 1 = 179 Hz f 3 = 1181 Hz 1 0.5
R 0 -0.5
-1 -1.5
0 f 2 = 582 Hz f 4 = 1830 Hz 5 10 t (ms) time 15 0 5 FFT(R) 0.15
10 15 20 25 0.1
30 35 582 frequency 1181 20 0.05
179 1830 0 0 500 1000 1500 frequency (Hz) 2000 2400 11 2500
5 20 10 15 y Dynamics of the Bat-Ball Collision AMN, AJP 68, 979-990 (2000) 20 20 2 z 2 EI 2 y z 2 A t 2 2 y F(z, t) 0 -5 y -10 z -15 1. Solve eigenvalue problem for normal modes of free bat (F=0) -20 0 10 15 20 25 30 35 5 modal frequencies and shapes y n (z),f n 2. Couple ball to bat via the ball-bat force F 3. Expand y in normal modes • Only modes with f n < ~2kHz matter 4. Solve coupled equations of motion for ball and bat (Runge-Kutta)
Vibrations, BBCOR, and the “Sweet Spot” 0.6
0.5
0.4
e 0.3
0.2
0.1
0
4 nodes 3 2 1
Ball-Bat COR 50 40 v f 30 20 E vib 10 vibrational energy 5 10 distance from tip (inches) 15 0 at ~ node 2
+
vibrations minimized COR maximized BBS maximized best “feel” 13
Vibrations and the ball-bat collision outside “sweet spot” 14
Vibrations and Broken Bats pitcher movie 0.000
5.000
10.000
15.000
20.000
catcher 25.000
30.000
35.000
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Independence of End Conditions • strike bat on barrel—look at movement in handle • handle moves only after ~0.6 ms delay • collision nearly over by then v (m/s) 30.00
• nothing on knob end matters • size, shape, hands, grip • boundary conditions • confirmed experimentally 20.00
10.00
0.00
-10.00
Batter could drop bat just before contact and it would have no effect on ball!!!
-20.00
-30.00
0 1 2 t (ms) 3 4 16 5
BBCOR and the Trampoline Effect (hollow bats)
The Ping!
Lowest Hoop (or wineglass) Mode
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The “Trampoline” Effect: A Simple Physical Picture • Two springs mutually compress • Energy shared between “ball spring” and “bat spring” Sharing depends on relative “stiffnesses” of springs • Energy stored in ball mostly dissipated (~80%!) • Energy stored 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, … demo 18
Forces on a Spinning Baseball in Flight
v •
Drag slows ball down
F = D 1 2 C D ˆ ω F M F D •
Magnus + mg deflects ball from straight line
F = M 1 2 C L mg
Runge-Kutta techniques used to solve eqns. of motion
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Consequences
• Drag – Fly balls don’t travel as far (factor of ~2!) – Pitched balls lose ~10% • Magnus – Movement on pitches (many examples later) – Batted balls • Backspin • Topspin longer fly balls; tricky popups nosedive on line drives; tricky grounders • Sidespin balls curve toward foul pole 20
New tools to study flight of baseball • PITCHf/x – Video tracking of pitched ball trajectory • HITf/x – Video tracking of initial batted ball trajectory • TrackMan – Doppler radar tracking of full pitched and batted ball trajectories • Hittracker – Careful observation of landing point and flight time of home runs 21
PITCHf/x and HITf/x
• Two video cameras @60 fps – “high home” and “high first” – tracks every pitch in every MLB ballpark • all data publicly available on web!
– tracks initial trajectory of batted ball • Used for analysis, TV broadcasts, MLB Gameday, etc.
Image, courtesy of Sportvision 22
Analyzing Batted Balls
Combining HITf/x with Hittracker – Initial position and velocity vectors, landing point r f =(x f ,y f ,z f ), and flight time (T) – Unknowns:
Example 1: Bonds record home run 24
Example 2: The “carry” of a fly ball
(379,20,5.2)
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Motivation: does the ball carry especially well in
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the new Yankee Stadium? “carry” ≡ (actual distance)/(vacuum distance) for same initial conditions
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HITf/x + hittracker Analysis: 4354 HR from 2009
Cleveland Denver Yankee Stadium
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An aside: Pitching at High Altitude Denver
7.5% 10% loss of velocity
Toronto
total movement 8”
Denver
12”
Toronto PITCHf/x data contain a wealth of information about drag and lift!
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TrackMan Data from StL, 2009
R vs. v 0 R vs.
0 USEFUL BENCHMARK 400 ft @ 103 mph ~5 ft per mph peaks @ 25 o -35 o
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What Constitutes a Well-Hit Ball?
BABIP V 0 >90 w/o home runs Basis for outcome HR independent batting metrics
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Pitched Ball Analysis: Using PITCHf/x to discover how pitchers do what they do
“Hitting is timing. Pitching is upsetting timing.”
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Ex 1: Mariano Rivera: Why is he so good?
?
Home Runs home plate Three Reasons: Location, Location, Location
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Ex 2: “Late Break”: Truth or Myth Mariano Rivera’s Cut Fastball View from above: actual trajectory ------- linear extrapolation - - - 32
Ex 3: A Pitcher’s Repertoire
4-seam fastball slider/cutter 2-seam fastball changeup curveball Catcher’s View
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Ex 4 Jon Lester vs. Brandon Webb
15 inches Brandon Webb is a “sinkerball” pitcher: Almost no rise on his fastball
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Ex 5 The Knuckleball
Tim Wakefield is a knuckleball pitcher: Chaotic Movement
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Now try to figure out this one…
In the video clip (4/29/11, TOR@NYY), look at… • spin axis – In what direction “should” the ball break?
• catcher’s glove – In what direction is the ball actually breaking?
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Stuff that keeps me busy
• Collision experiments & calculations to elucidate trampoline effect • More studies of baseball trajectories • Careful studies of PITCHf/x cameras and sources of systematic error • Experiments on high-speed oblique collisions – To quantify spin on batted ball 37
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.illinois.edu/physicsofbaseball – [email protected]
• I am living proof that knowing the physics doesn’t help you play the game better!
@ Red Sox Fantasy Camp, Feb. 1-7, 2009
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