1 Physics and Baseball: Having Your Cake and Eating it Too Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): ; Oct 2002 Alan M. Nathan Department of Physics University of Illinois

2 How Does a Physicist Analyze the Game of Baseball? October 27, 2004: the day the curse was broken Having Your Cake and Eating it Too

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

4 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.”

5 Outline How does a baseball bat work? The flight of a baseball. Leaving the no-spin zone. Putting it all together.

6 “You can observe a lot by watching” Champaign News-Gazette CE Composites --Yogi Berra Easton Sports

7 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 ( ) –lots of backspin

8 BBS = 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  BBS  0.2 v ball v bat Kinematics of Ball-Bat Collision v ball v bat BBS

9 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) q=0.20 BBS = q v ball + (1+q) v bat v ball v bat BBS

10 Collision Efficiency q Can Be Measured Air cannon fires ball onto stationary bat q = v out /v in Used by NCAA, ASA, … to regulate/limit performance of bats Sports Sciences WSU

11 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

12 Modal Analysis of a Baseball Bat frequency time f 1 = 179 Hz f 2 = 582 Hz f 3 = 1181 Hz f 4 = 1830 Hz

13 Vibrations, COR, and the “Sweet Spot” E vib vfvf e + Strike bat here best performance & ~ node 2

14 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 Independence of End Conditions

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

16 Vibrations and Broken Bats movie pitcher catcher

17 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” Does Aluminum Outperform Wood? demo

18 A Closer Look at Hoop Modes “hoop” modes: cos(2  ) “ping” Thanks to Dan Russell

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

20 Softball Data and Model Conclusion: essential physics understood Russell, Smith, AMN

21 Does Aluminum Outperform Wood? YES!

22 Forces on a Baseball in Flight Gravity Drag (“air resistance”) Lift (or “Magnus”) mg FdFd FLFL  Courtesy, Popular Mechanics F d =½ C D  Av 2 F L = ½ C L  Av 2 direction leading edge is turning

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

24 Effect of Drag and Lift on Trajectories drag effect is huge lift effect is smaller but significant mg FdFd F L (Magnus) 

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 ~30 o -35 o

26 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.

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

28 How Far Did That Home Run Travel? Ball leaves bat Hits stands D from home plate, H above ground How far would it have gone if no obstruction?

ft/30 ft Range= Time can resolve 4 s 5 s 7 s See

30 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?

31 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

32 Undercutting the ball  backspin Ball10 0 downward Bat 10 0 upward D = center-to-center offset trajectories “vertical sweet spot” What’s going on here??

33 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?

34 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

35 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?

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

37 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 Thanks for your attention and go Red Sox!