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The Physics of Hitting a Home Run

Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002. The Physics of Hitting a Home Run. Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu. 1927 Yankees: Greatest baseball team ever assembled.

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The Physics of Hitting a Home Run

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

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

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

  5. “You can observe a lot by watching” Champaign News-Gazette --Yogi Berra Easton Sports CEComposites 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 (300-350) • lots of backspin UBC Colloquium 10/5/06

  7. vball vbat vf Kinematics of Ball-Bat Collision vf = 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 •  vf 0.2 vball + 1.2 vbat Conclusion: vbat matters much more than vball UBC Colloquium 10/5/06

  8. vball vbat vf q=0.20 Kinematics of Ball-Bat Collision • r = mball /Mbat,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) UBC Colloquium 10/5/06

  9. Collision Efficiency q Can Be Measured • Air cannon to fire ball onto stationary bat • q = vout/vin • Used by NCAA, ASA, … to regulate/limit performance of bats Sports Sciences Lab @ WSU 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, vf • Dynamic model of collision • Treat bat as nonuniform beam • Treat ball as damped spring UBC Colloquium 10/5/06

  11. 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 UBC Colloquium 10/5/06

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

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

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

  16. 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” • Also seen in golf, tennis, … UBC Colloquium 10/5/06

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

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

  19. The Trampoline Effect: A Closer Look Bending Modes vs. Shell Modes • k  R4: 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 to optimize…. kbat//kball small and fhoop > 1 UBC Colloquium 10/5/06

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

  21. FL(Magnus)  Drag: Fd = ½ CDAv2 “Magnus” or “Lift”: FL= ½ CLAv2 Fd mg Aerodynamics of a Baseball (in direction leading edge is turning) CD~ 0.2-0.5 CL ~ R/v UBC Colloquium 10/5/06

  22. Effect of Drag and Lift on Trajectories FL(Magnus)  Fd mg • drag effect is huge • lift effect is smaller but significant 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 ~30o-35o UBC Colloquium 10/5/06

  24. FL(Magnus)  Fd mg Some Effects of Lift • 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. 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 x0,y0,vx,vy,,ay • Use ay to determine Magnus force as function ofv,  UBC Colloquium 10/5/06

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

  27. Typical Motion Capture Datameasure spin, CM trajectory CM trajectory Note: topspin  ay > g UBC Colloquium 10/5/06

  28. Results for Lift Coefficient CL FL= 1/2ACLv2 S=r/v 100 mph, 2000 rpm S=0.17 Conclusion: data qualitatively consistent (~20%) 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? 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 UBC Colloquium 10/5/06

  31. Ball100 downward D = center-to-center offset Bat 100 upward Undercutting the ball  backspin trajectories “vertical sweet spot” UBC Colloquium 10/5/06

  32. 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 UBC Colloquium 10/5/06

  33. 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 UBC Colloquium 10/5/06

  34. 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! 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! 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! UBC Colloquium 10/5/06

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