1 Baseball and Mathematics: It’s More Than Batting Averages ---Alan Nathan

Solvay Conference: Greatest physics team ever assembled The Baseball/Physics Connection 1927 Yankees: Greatest baseball team ever assembled MVP’s

3 A good book to read…. Prof. Bob Adair

4 Another very useful reference…

5 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

6 “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

7 pitch speed bat speed “collision efficiency”: a property of the ball and bat BBS = q v pitch + (1+q) v bat typical numbers: q = q = 1.2 example: 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 What Determines Batted Ball Speed?

8 The simple stuff: kinematics 1. m/M eff = ball mass/effective bat mass  0.25 bat recoil 2. e = elasticity of collision  0.50 “ball-bat coefficient of restitution” (BBCOR)  harder stuff: dynamics! 3. For m/M eff <<1 and e  1, q  1 BBS = q vpitch + (1+q) vbat

9 ball bat Mass= Dynamics of Ball-Bat Collision: A Toy Model  >> 1 m on M a +M b (1 on 6)  << 1 m on M a (1 on 2) rigid flexible q

10 The Ball-Bat Force (from real data) Vibrational modes with f<2 kHz are important

11 Experimental Studies of Bat Vibrations frequency time f 1 = 179 Hz f 2 = 582 Hz f 3 = 1181 Hz f 4 = 1830 Hz

12 20 Dynamics of the Bat-Ball Collision AMN, AJP 68, (2000) 1.Solve eigenvalue problem for normal modes of free bat (F=0)  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)  BBS, recoil of bat, energy dissipation in ball and bat (BBCOR) 20 y y z

13 Vibrations, BBCOR, and the “Sweet Spot” E vib vfvf e + at ~ node 2 vibrations minimized COR maximized BBS maximized best “feel”

14 Vibrations and the ball-bat collision outside“sweet spot”

15 Vibrations and Broken Bats movie pitcher catcher

16 strike bat on barrel—look at movement in handle handle moves only after ~0.6 ms delay collision nearly over by then nothing on knob end matters size, shape, hands, grip boundary conditions confirmed experimentally Independence of End Conditions Batter could drop bat just before contact and it would have no effect on ball!!!

17 BBCOR and the Trampoline Effect (hollow bats) The Ping! Lowest Hoop (or wineglass) Mode

18 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, … The “Trampoline” Effect: A Simple Physical Picture demo

19 Forces on a Spinning Baseball in Flight mg FDFD FMFM Drag slows ball down Magnus + mg deflects ball from straight line Runge-Kutta techniques used to solve eqns. of motion

20 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  longer fly balls; tricky popups Topspin  nosedive on line drives; tricky grounders Sidespin  balls curve toward foul pole

21 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

22 PITCHf/x and HITf/x Two video 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 Marv White, Physics, UIUC, 1969 Marv White, Physics, UIUC, 1969

23 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:,  b,  s –Use non-linear least-squares fitting to fit to r f (T) Get the full trajectory Amazingly robust

24 Example 1: Bonds record home run

25 Example 2: The “carry” of a fly ball Motivation: does the ball carry especially well in the new Yankee Stadium? “carry” ≡ (actual distance)/(vacuum distance) for same initial conditions (379,20,5.2)

26 HITf/x + hittracker Analysis: 4354 HR from 2009 Denver ClevelandYankee Stadium

27 An aside: Pitching at High Altitude 10% loss of velocity total movement 12” 7.5% 8” PITCHf/x data contain a wealth of information about drag and lift! Toronto Denver

28 TrackMan Data from StL, 2009 R vs. v 0 R vs.  0 USEFUL BENCHMARK mph ~5 ft per mph 25 o -35 o

29 What Constitutes a Well-Hit Ball? w/o home runs HR BABIP V 0 >90 Basis for outcome- independent batting metrics

30 Pitched Ball Analysis: Using PITCHf/x to discover how pitchers do what they do “Hitting is timing. Pitching is upsetting timing.”

31 home plate Ex 1: Mariano Rivera: Why is he so good? ? Three Reasons: Location, Location, Location Home Runs

32 Ex 2: “Late Break”: Truth or Myth Mariano Rivera’s Cut Fastball View from above: actual trajectory linear extrapolation

33 Ex 3: A Pitcher’s Repertoire Catcher’s View 4-seam fastball 2-seam fastball changeup curveball slider/cutter

34 Ex 4 Jon Lester vs. Brandon Webb Brandon Webb is a “sinkerball” pitcher: Almost no rise on his fastball 15 inches

35 Ex 5 The Knuckleball Tim Wakefield is a knuckleball pitcher: Chaotic Movement

36 Now try to figure out this one… In the video clip (4/29/11, look at… spin axis –In what direction “should” the ball break? catcher’s glove –In what direction is the ball actually breaking?

37 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

38 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 I am living proof that knowing the physics doesn’t help you play the game Red Sox Fantasy Camp, Feb. 1-7, 2009