How a Physicist Analyzes the Game of Baseball

1. How a Physicist Analyzes the Game of Baseball Alan M. Nathan a-nathan@uiuc.edu webusers.npl.uiuc.edu/~a-nathan/pob Department of Physics University of Illinois

2. 1927 Yankees: Greatest baseball team ever assembled 1927 Solvay Conference: Greatest physics team ever assembled MVP’s Baseball and Physics

3. 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.”

4. Some Topics I Will Cover • How does a baseball bat work? • The flight of a baseball • Leaving the no-spin zone • Putting it all together

5. Champaign News-Gazette “You can observe a lot by watching” --Yogi Berra Easton Sports CEComposites

6. Description of Ball-Bat Collision • forces large, time short • >8000 lbs, <1 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 batted ball speed • 100 mph~400 ft, each additional mph ~ 5-6’ • optimum take-off angle (300-350) • lots of backspin

7. vball vbat BBS Kinematics of Ball-Bat Collision BBS = q vball + (1+q) vbat • 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 Conclusion: vbat matters much more than vball

8. Kinematics of Ball-Bat Collision vball vbat BBS q=0.20 BBS = q vball + (1+q) vbat • e:“coefficient of restitution” 0.50 • (energy dissipation—mainly in ball, some in bat) • r = mball /Mbat,eff :bat recoil factor = 0.25 • (momentum and angular momentum conservation) • ---heavier is better but…

9. Batting cage study show how bat speed depends on MOI for college/semipro baseball players aluminum wood

10. 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

11. Regulating Performance of Non-Wood Bats (NCAA) • Specify maximum q (“BESR”=q+1/2) • relative to wood • implies bats swung alike will perform alike • Specify minimum MOI to limit bat speed • smaller than wood • Together, these determine a maximum BBS • gap between wood and aluminum  5% BBS = q vball + (1+q) vbat

12. BESR MOI aluminum -5 rule

13. 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, BBS • Dynamic model of collision • Treat bat as nonuniform beam • Treat ball as damped spring

14. f1 = 179 Hz f3 = 1181 Hz f2 = 582 Hz f4 = 1830 Hz frequency time Modal Analysis of a Baseball Bat www.kettering.edu/~drussell/bats.html

15. Vibrations, COR, and the “Sweet Spot” Strike bat here + best performance & feel @ ~ node 2 e vf Evib

16. Independence of End Conditions • 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

17. q independent of end conditions: experimental proof Conclusion: mass added in knob has no effect on collision efficiency (q)

18. Vibrations and Broken Bats pitcher catcher movie

19. Why Is Aluminum Better Than Wood? 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” demo

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”—confirmed by experiment • …and higher BBS • Also seen in golf, tennis, … demo

21. FM  Fd Fd=½ CDAv2 mg FM= ½ CLAv2 Courtesy, Popular Mechanics direction leading edge is turning Aerodynamics of a Baseball • Gravity • Drag (“air resistance”) • Lift (or “Magnus”)

22. drag/wt0.8 Magnus/wt=0.58 Measuring drag and Magnus forces by high-speed tracking Magnus force is much easier to measure than drag force

23. “Drag crisis?” Typical values of drag and lift

24. Effect of Drag and Lift on Trajectories FM  Fd mg • drag effect is huge • lift effect is smaller but significant

25. 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

26. FM  Fd mg Some Effects of 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.

27. The PITCHf/x Tracking SystemA New Tool to Study Baseball Flight

28. How Does PITCHf/x Work? • Two video cameras track baseball in 1/60-sec intervals • usually “high home” and “high first” • third CF camera used establises ht. of strike zone for each batter • Pattern-recognition software to identify “blobs” • Camera calibration to convert pixels to (x,y,z) 9-parameter fit to trajectory • constant acceleration for x(t),y(t),z(t) • Use fit to calculate lots of stuff • The full trajectory • The “break” • Drag and Magnus forces

29. Pitch ClassificationJon Lester, Aug 3, 2007 @ Seattle LHP Catcher’s View I: Nearly overhand fastball II: Cut Fastball III: ¾ Fastball IV: Curveball

30. Courtesy, Ryutaro Himeno Courtesy, The New York TImes Daisuke Matsuzaka: Does he or doesn’t he? What’s the Deal with the Gyroball?

32. Barry Bond’s 756th Home Run • PITCHf/x data tracked hit ball over first 20 ft • Precision measurement of endpoint and time • Inferred: v0=112 mph; =27o up; =16o to right of dead center; =1186 rpm (backspin) and 189 rpm (sidespin, breaking to center)

33. 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?

34. 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

35. Ball100 downward D = center-to-center offset Bat 100 upward What’s going on here?? Undercutting the ball  backspin trajectories “vertical sweet spot”

37. Another familiar result: bat hits under ball: popup to opposite field bat tilted downward Catcher’s View bat hits over ball: grounder to pull field

38. Putting it all Together:Can curveball be hit farther than fastball? • Bat-Ball Collision Dynamics • A fastball will be hit faster • A curveball will be hit with more backspin

39. Fastball with backspin Curveball: spin doesn’t reverse Curveball with topspin curveball can be hit with more backspin: WHY? Fastball: spin must reverse Net effect: backspin larger for curveball

40. 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!

41. Work in Progress • Collision experiments & calculations to elucidate trampoline effect • New studies of aerodynamics using Doppler radar • Experiments on high-speed oblique collisions • A book, with Aussi Rod Cross

42. 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 • a-nathan@uiuc.edu • Thanks for your attention and go Red Sox!