1. UBC Colloquium 10/5/06 1 Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002 Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu The Physics of Hitting a Home Run

2. UBC Colloquium 10/5/06 2 1927 Solvay Conference: Greatest physics team ever assembled Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVP’s

3. UBC Colloquium 10/5/06 3 Adair’s Book: An Excellent Reference “Our goal is not to reform the game but to understand it. “The physicist’s model of the game must fit the game.”

4. UBC Colloquium 10/5/06 4 1.How does a baseball bat work? 2.Aerodynamics: flight of a baseball 3.Leaving the no-spin zone 4.Putting it all together The Physics of Hitting a Home Run

5. UBC Colloquium 10/5/06 5 “You can observe a lot by watching” Champaign News-Gazette CE Composites --Yogi Berra Easton Sports

6. UBC Colloquium 10/5/06 6 Brief 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 hit ball speed (100 mph  ~400 ft) –optimum take-off angle (30 0 -35 0 ) –lots of backspin

7. UBC Colloquium 10/5/06 7 v f = q v ball + (1+q) v bat Conclusion: v bat matters much more than v ball q  “Collision Efficiency” Joint property of ball & bat  independent of reference frame  ~independent of “end conditions”—more later  weakly dependent on v rel Superball-wall: q  1 Ball-Bat near “sweet spot”: q  0.2  v f  0.2 v ball + 1.2 v bat v ball v bat vfvf Kinematics of Ball-Bat Collision

8. UBC Colloquium 10/5/06 8 Kinematics of Ball-Bat Collision r = m ball /M bat,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) v ball v bat vfvf q=0.20

9. UBC Colloquium 10/5/06 9 Collision Efficiency q Can Be Measured Air cannon to fire ball onto stationary bat q = v out /v in Used by NCAA, ASA, … to regulate/limit performance of bats Sports Sciences Lab @ WSU

10. UBC Colloquium 10/5/06 10 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, v f Dynamic model of collision –Treat bat as nonuniform beam –Treat ball as damped spring

11. UBC Colloquium 10/5/06 11 Modal Analysis of a Baseball Bat www.kettering.edu/~drussell/bats.html frequency time f 1 = 179 Hz f 2 = 582 Hz f 3 = 1181 Hz f 4 = 1830 Hz

12. UBC Colloquium 10/5/06 12 Vibrations, COR, and the “Sweet Spot” E vib vfvf e Node of 1 nd mode + Strike bat here Measure response here

13. UBC Colloquium 10/5/06 13 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 Independence of End Conditions

14. UBC Colloquium 10/5/06 14 q independent of end conditions: experimental proof Conclusion: mass added in knob has no effect on collision efficiency (q)

15. UBC Colloquium 10/5/06 15 Aluminum has thin shell Hoop modes give “trampoline” effect –larger COR, v f Why Does Aluminum Outperform Wood?

16. UBC Colloquium 10/5/06 16 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, … The “Trampoline” Effect: A Simple Physical Picture

17. UBC Colloquium 10/5/06 17 The Trampoline Effect: A Closer Look “hoop” modes: cos(2  ) “ping” Thanks to Dan Russell

18. UBC Colloquium 10/5/06 18 Wood vs. Aluminum: Where Does the Energy Go?

19. UBC Colloquium 10/5/06 19 The Trampoline Effect: A Closer Look to optimize…. k bat/ /k ball small and f hoop  > 1 k  R 4 : large in barrel  little energy stored f (170 Hz, etc) > 1/   energy goes into vibrations k  (t/R) 3 : small in barrel  more energy stored f (2-3 kHz) < 1/   energy mostly restored Bending Modes vs. Shell Modes

20. UBC Colloquium 10/5/06 20 Softball Data and Model essential physics understood

21. UBC Colloquium 10/5/06 21 Aerodynamics of a Baseball Drag: F d = ½ C D  Av 2 “Magnus” or “Lift”: F L = ½ C L  Av 2 mg FdFd F L (Magnus)  C D ~ 0.2-0.5 C L ~ R  /v (in direction leading edge is turning)

22. UBC Colloquium 10/5/06 22 Effect of Drag and Lift on Trajectories drag effect is huge lift effect is smaller but significant mg FdFd F L (Magnus) 

23. UBC Colloquium 10/5/06 23 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 ~30 o -35 o

24. UBC Colloquium 10/5/06 24 Some Effects of Lift mg FdFd F L (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.

25. UBC Colloquium 10/5/06 25 New Experiment at Illinois Fire baseball horizontally from pitching machine Use motion capture to track ball over ~5m of flight and determine x 0,y 0,v x,v y, ,a y Use a y to determine Magnus force as function of v, 

26. UBC Colloquium 10/5/06 26 Motion Capture Experiment Joe Hopkins, Lance Chong, Hank Kaczmarski, AMN Two-wheel pitching machine Baseball with reflecting dot Motion Capture System

27. UBC Colloquium 10/5/06 27 Typical Motion Capture Data measure spin, CM trajectory Note: topspin  a y > g CM trajectory

28. UBC Colloquium 10/5/06 28 Results for Lift Coefficient C L F L = 1/2  AC L v 2 S=r  /v 100 mph, 2000 rpm  S=0.17 Conclusion: data qualitatively consistent (~20%)

29. UBC Colloquium 10/5/06 29 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?

30. UBC Colloquium 10/5/06 30 Oblique Collisions: Leaving the No-Spin Zone Oblique  friction  spin transverse velocity reduced spin increased 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

31. UBC Colloquium 10/5/06 31 Undercutting the ball  backspin Ball10 0 downward Bat 10 0 upward D = center-to-center offset trajectories “vertical sweet spot”

32. UBC Colloquium 10/5/06 32 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?

33. UBC Colloquium 10/5/06 33 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

34. UBC Colloquium 10/5/06 34 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?

35. UBC Colloquium 10/5/06 35 Work in Progress Collision experiments & calculations to elucidate trampoline effect New studies of aerodynamics Experiments on oblique collisions –No data!

36. UBC Colloquium 10/5/06 36 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 –www.npl.uiuc.edu/~a-nathan/pob –a-nathan@uiuc.edu Go Red Sox!