The Physics of Hitting a Home Run Alan M. Nathan,University of Illinois www.npl.uiuc.edu/~a-nathan/pob a-nathan @uiuc.edu Thanks to J. J. Crisco & R. M. Greenwald Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002 Computer rendition of the swinging bat, the incoming and outgoing ball, and the collision between them, from a study in which high-speed video was used to track the location of the ball and several points on the bat as the batter swung at a ball delivered by a pitching machine. The separation between adjacent frames is 4 ms. Data from images such as these can be used to determine the relationship between the outgoing speed, incoming speed, and bat speed, as a function of impact location along the bat. For example, the above image clearly shows that the outgoing ball, shown as the upper set of images, has a higher speed than the incoming ball, shown as the lower set of images. The purple dot shows the evolution of the instantaneous rotation axis of the bat. Interestingly, just prior to the collision, the rotation axis approximately coincides with the wrist of the lower hand. UW Colloquium 10/31/05 1

Greatest baseball team Baseball and Physics 1927 Yankees: Greatest baseball team ever assembled MVP’s 1927 Solvay Conference: Greatest physics team ever assembled UW Colloquium 10/31/05

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

Hitting and Pitching, Thinking and Guessing Greg Maddox: every pitch looks alike for the first 20 ft or so. Graphic courtesy of Bob Adair and NYT UW Colloquium 10/31/05

Example: Tim Wakefield’s Knuckleball UW Colloquium 10/31/05

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

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 UW Colloquium 10/31/05

Kinematics of Ball-Bat Collision vball vbat vf vf = q vball + (1+q) vbat q  “Collision Efficiency” 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 Primary dependence of vf on ball, bat speeds is in the relationship…weak dependence of ea itself. Conclusion: vbat matters much more than vball UW Colloquium 10/31/05

Kinematics of Ball-Bat Collision vball vbat vf r = mball /Mbat,eff : bat recoil factor =  0.25 (momentum and angular momentum conservation) e: “coefficient of restitution”  0.50 (energy dissipation—mainly in ball, some in bat) q=0.20 Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e, which depends on rel ball-bat speed. Mass is effective mass due to extended nature of bat. Inefficient collision (1 for superball on rigid wall) Bat speed matters much more than ball speed UW Colloquium 10/31/05

Kinematics of Ball-Bat Collision r = mball /Mbat,eff: bat recoil factor =  0.25 (momentum and angular momentum conservation) heavier bat better but… Primary dependence of vf on ball, bat speeds is in the relationship…weaker dependence through e, which depends on rel ball-bat speed. Mass is effective mass due to extended nature of bat. Inefficient collision (1 for superball on rigid wall) Bat speed matters much more than ball speed UW Colloquium 10/31/05

The Ideal Bat Weight or Iknob Experiments: knob ~ (1/Iknob)0.3 NB: Data suggest Iknob matters much more than weight per se. Observation: Batters prefer lighter bats UW Colloquium 10/31/05

Dynamic Model for Ball-Bat Collision 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 vf So far, just kinematic, plus a phenomonlogoical treatment of energy losses via COR. Now we want to dissect the collision process, time slice by time slice, to see what is really going on during the time the ball and bat are in contact. In doing so, we want to try to do a strict accounting for where the energy goes in the collision. So, we want to go beyond kinematics and talk about dynamics. We know that a purely rigid body treatment cannot be right…for example, we know that the collision can excite vibrations in the bat. The swing: Early—translation Later—rotation The hit Bat slows down, vibrates UW Colloquium 10/31/05

The Details: A Dynamic Model 20 y z 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 Cover only briefly UW Colloquium 10/31/05

Modal Analysis of a Baseball Bat www.kettering.edu/~drussell/bats.html f1 = 179 Hz f2 = 582 Hz f3 = 1181 Hz f4 = 1830 Hz time frequency Note: nodes stack up at barrel end-->”sweet spot zone” UW Colloquium 10/31/05

Some Interesting Insights: Bat Recoil, Vibrations, COR, and “Sweet Spot” Node of 1nd mode + e vf Evib Impact at various places on barrel, motion at handle near middle of top hand. Players say that is where the sting occurs. Bottom hand not as sensitive because of damping due to top hand. Happens to nearly coincide with node of lowest mode. COR vs z feel 1. Vibrations No vibrations of 1st mode seen at handle node Impact at node of 2nd mode: only high-freq. vibration. At other locations, 560 Hz mode there. That is probably the buzz. Hands not sensitive to higher frequencies. 2. Maximum of e, max. of vf = minimum of vibrational energy at 2nd node ~ 1 ms  only lowest 4 modes excited UW Colloquium 10/31/05

Experimental Data: Dependence of COR on Impact Location ball incident on bat at rest For rigid, peak is at CM For full calculation, peak determined by interplay between rotational recoil and vibrational nodes. At lowest node, rigid-full Conclusion: essential physics under control UW Colloquium 10/31/05

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 UW Colloquium 10/31/05

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

Why Does Aluminum Outperform Wood? 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  This summarizes the important properties of bats UW Colloquium 10/31/05

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, … Ask question: Which give more “power”: tighter strings or looser strings on tennis racket? Then ask (if they get it wrong): can a person bounce higher from a hardwood floor or from a trampoline? UW Colloquium 10/31/05

The Trampoline Effect: A Closer Look “hoop” modes: cos(2) k  (t/R)3: hoop mode largest in barrel f2 (1-3 kHz) < 1/   1kHz  energy mostly restored (unlike bending modes) Thanks to Dan Russell The f-tau relationship is important…the reason why there is no trampoline effect from bending modes--see next slide Tricks: double wall bats; composite bats “ping” UW Colloquium 10/31/05

essential physics understood Data and Model to optimize…. kbat small fhoop > 1 essential physics understood UW Colloquium 10/31/05

Effect of Spin on Baseball Trajectory mg Fd FL (Magnus)  Drag: Fd = ½ CDAv2 “Magnus” or “Lift”: FL = ½ CLAv2 (in direction leading edge is turning) CD~ 0.2-0.5 CL ~ R/v UW Colloquium 10/31/05

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 of v,  UW Colloquium 10/31/05

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

Experiment: Sample MoCap Data y z topspin  ay > g y = ½ ayt2 work in progress UW Colloquium 10/31/05

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

Oblique Collisions: Leaving the No-Spin Zone Friction … sliding/rolling vs. gripping transverse velocity reduced, spin increased vT′ ~ 5/7 vT  ~ vT′/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 UW Colloquium 10/31/05

Undercutting the ball  backspin Ball100 downward Bat 100 upward D = center-to-center offset trajectories Undercut too much: popup with lots of backwpin Undercut too little: low trajectory with little backspin UW Colloquium 10/31/05

Fastball: spin reverses Curveball: spin doesn’t reverse   larger for curveball Undercut too much: popup with lots of backwpin Undercut too little: low trajectory with little backspin More friction does not help UW Colloquium 10/31/05

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! UW Colloquium 10/31/05 1

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 UW Colloquium 10/31/05

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! UW Colloquium 10/31/05