1. UW Colloquium 10/31/05 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. UW Colloquium 10/31/05 2 1927 Solvay Conference: Greatest physics team ever assembled Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVPs

3. UW Colloquium 10/31/05 3 Philosophical Remarks: (Courtesy of Bob Adair) …the physics of baseball is not the clean, well-defined physics of fundamental matters. Hence conclusions must depend on approximations and estimates. But estimates are part of the physicists repertoire... The physicists model of the game must fit the game. Our goal is not to reform the game but to understand it.

4. UW Colloquium 10/31/05 4 Hitting is timing; pitching is upsetting timing Hitting the Baseball: the most difficult feat in sports Hitting is fifty percent above the shoulders 1955 Topps cards from my personal collection

5. UW Colloquium 10/31/05 5 Graphic courtesy of Bob Adair and NYT Hitting and Pitching, Thinking and Guessing

6. UW Colloquium 10/31/05 6 Example: Tim Wakefields Knuckleball

7. UW Colloquium 10/31/05 7 1.How does a baseball bat work? 2.Why does aluminum outperform wood? 3.How does spin affect flight of baseball? 4.Can a curveball be hit farther than a fastball? The Physics of Hitting a Home Run

8. UW Colloquium 10/31/05 8 Brief Description of Ball-Bat Collision forces large, time short – >8000 lbs, <1 ms ball compresses, stops, expands – KE PE KE – bat bends & compresses lots of energy dissipated (COR) – distortion of ball – vibrations in bat to hit home run…. –large hit ball speed –optimum take-off angle –lots of backspin Courtesy of CE Composites

9. UW Colloquium 10/31/05 9 v f = q v ball + (1+q) v bat Conclusion: v bat matters much more than v ball q Collision Efficiency property of ball & bat independent of reference frame ~independent of end conditionsmore 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

10. UW Colloquium 10/31/05 10 Kinematics of Ball-Bat Collision r = m ball /M bat,eff : bat recoil factor = 0.25 (momentum and angular momentum conservation) e: coefficient of restitution 0.50 (energy dissipationmainly in ball, some in bat) v ball v bat vfvf q=0.20

11. UW Colloquium 10/31/05 11 Kinematics of Ball-Bat Collision r = m ball /M bat,eff : bat recoil factor = 0.25 (momentum and angular momentum conservation) heavier bat better but…

12. UW Colloquium 10/31/05 12 The Ideal Bat Weight or I knob Observation: Batters prefer lighter bats Experiments: knob ~ (1/I knob ) 0.3

13. UW Colloquium 10/31/05 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 lower v f

14. UW Colloquium 10/31/05 14 The Details: A Dynamic Model 20 y z y Step 1: Solve eigenvalue problem for free vibrations Step 2: Nonlinear lossy spring for ball-bat interaction F(t) Step 3: Expand in normal modes and solve

15. UW Colloquium 10/31/05 15 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

16. UW Colloquium 10/31/05 16 Some Interesting Insights: Bat Recoil, Vibrations, COR, and Sweet Spot E vib vfvf e Node of 1 nd mode + ~ 1 ms only lowest 4 modes excited

17. UW Colloquium 10/31/05 17 Experimental Data: Dependence of COR on Impact Location ball incident on bat at rest Conclusion: essential physics under control

18. UW Colloquium 10/31/05 18 handle moves only after ~0.6 ms delay collision nearly over by then nothing on knob end matters size, shape boundary conditions hands Independence of End Conditions

19. UW Colloquium 10/31/05 19 pitcher catcher Vibrations and Broken Bats outside inside node

20. UW Colloquium 10/31/05 20 Aluminum has thin shell –Less mass in barrel –easier to swing and control –but less effective at transferring energy –for many bats cancels –Hoop modes –trampoline effect –larger COR Why Does Aluminum Outperform Wood?

21. UW Colloquium 10/31/05 21 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

22. UW Colloquium 10/31/05 22 The Trampoline Effect: A Closer Look hoop modes: cos(2 ) k (t/R) 3 : hoop mode largest in barrel f 2 (1-3 kHz) < 1/ 1kHz energy mostly restored (unlike bending modes) ping Thanks to Dan Russell

23. UW Colloquium 10/31/05 23 Data and Model to optimize…. k bat small f hoop > 1 essential physics understood

24. UW Colloquium 10/31/05 24 Effect of Spin on Baseball Trajectory 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)

25. UW Colloquium 10/31/05 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. UW Colloquium 10/31/05 26 Motion Capture Experiment Joe Hopkins, Lance Chong, Hank Kaczmarski, AMN Two-wheel pitching machine Baseball with reflecting dot Motion Capture System

27. UW Colloquium 10/31/05 27 Experiment: Sample MoCap Data y z topspin a y > g y = ½ a y t 2 work in progress

28. UW Colloquium 10/31/05 28 Some Typical Results Lift … --increases range --reduces optimum angle

29. UW Colloquium 10/31/05 29 Oblique Collisions: Leaving the No-Spin Zone Friction … sliding/rolling vs. gripping transverse velocity reduced, spin increased v T ~ 5/7 v T ~ v T /R Familiar Results Balls hit to left/right break toward foul line Topspin gives tricky bounces in infield Pop fouls behind the plate curve back toward field Backspin keeps fly ball in air longer f

30. UW Colloquium 10/31/05 30 Undercutting the ball backspin Ball10 0 downward Bat 10 0 upward D = center-to-center offset trajectories

31. UW Colloquium 10/31/05 31 larger for curveball Fastball: spin reverses Curveball: spin doesnt reverse

32. UW Colloquium 10/31/05 32 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?

33. UW Colloquium 10/31/05 33 Work in Progress Collision experiments & calculations to elucidate trampoline effect New measurements of lift and drag Experiments on oblique collisions –Rod Cross & AMN: rolling almost works at low speed –AMN: studies in progress at high speed

34. UW Colloquium 10/31/05 34 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!