Baseball: It’s Not Nuclear Physics (or is it. ) Alan M Baseball: It’s Not Nuclear Physics (or is it?!) Alan M. Nathan University of Illinois GWU Colloquium, October 21, 1999 Introduction Hitting the Baseball The Flight of the Baseball Pitching the Baseball Summary I’ve been doing physics all my professional life but I’ve been involved in one way or another with baseball for even longer. Only recently have I taken an interest in the physics of the game. The goal is not to try to improve the game. The game itself has evolved and been perfected over more than a century…if a ballplayer says that a particular thing works, then it probably does. They have figured out how to optimize things. Our goal is to try to make physical sense of the whole thing. The game itself is our laboratory…we observe, construct a model, use physics principles and “expt” to constrain the model, predict, then iterate. In fact it has proved to be very satisfying that, although there is not much that we canunderstand from "first principles", it is possible to create sensible models for many familiar aspects of the game, such as the flight of the baseball, the collision between the ball and bat, etc. To the extent that our models reproduce what we observe, then we can be happy that we understand the underlying physics. In that sense, baseball physics is very similar to nuclear physics, my "day job". There are other similarities as well, that I will try to point out as we go along. 1

REFERENCES The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN 0-06-096461-8 The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN 0-88318-946-1 www.physics.usyd.edu.au/~cross L. L. Van Zandt, AJP 60, 72 (1991) www.npl.uiuc.edu/~a-nathan Introduction: Adair book very nice. Lots of interesting facts about the game. Good insight as to how a physicist thinks about the world around him. Intellectually honest: tells you what we know and understand very well, what we can make an educated guess about, and what we don’t even have a clue about.

“...the most difficult thing to do in sports” Hitting the Baseball “...the most difficult thing to do in sports” --Ted Williams BA: .344 SA: .634 OBP: .483 HR: 521 90 mph ball…<0.5 s to evaluate, guess at trajectory, go through a complicated physiological process to transfer energy from your muscles to the bat, and put the bat in the right place at the right time. Very easy to fail! E.g., if off 0.01 sec in timing, ball moves 1’. Compare with golf, where ball is not even moving!

Speed of Hit Ball: What does it depend on? Speed is important: 105 mph gives ~400 ft each mph is worth 5 ft The basic stuff (“kinematics”) speed of pitched ball speed of bat weight of bat The really interesting stuff (“dynamics”) “bounciness” of ball and bat weight distribution of bat vibrations of bat HOLD BAT & BALL Broad subject Focus on 1 ms during actual collision What physics can we bring to bear …or, what aspects of collision lead to well-hit ball If you want to optimize your chances of getting a hit, you want to maximize the speed of the hit ball. whether swing for fences or line drive through infield neglect place hitting, which is another art by itself

What Determines Batted Ball Speed? 1. pitched ball speed 2. bat speed Rigid-Body Kinematics: V = 0.25 Vball + 1.25 Vbat Formula assumes typical bat, ball parameters Bat speed matter more…should be no surprise (issue is momentum) Can hit a homer from lob but not by bunting! Conclusion: Bat Speed Matters More!

What Determines Batted Ball Speed? 3. Mass of bat larger mass  lower bat speed bat speed vs mass 1. Kinematics: For given bat speed, heavy bat give higher ball speed than light bat 2. Kinematics: For given bat KE, light bat gives higher ball speed than heavy bat 3. These, plus common sense that batter can swing light bat faster than heavier bat, establish limiting curves 4. Model: fixed energy (force x distance), shared by bat and body (about equally for “normal” bat). Result: as mbat increases , Ebat increases, Ebody decreases, and vbat decreases (approx as 1/m, in agreement with “experiment”) 5. Tendency towards lighter bats. NCAA rules, etc.. Seems to contradict this analysis. Mass distribution not considered here (impt for wood-Al differences). Also control, timing, and reaction time. ball speed vs mass Conclusion: mass of bat matters….but not a lot

What Determines Batted Ball Speed? 4. Inelasticity Ball compresses kinetic energy stored in “spring” Ball expands kinetic energy restored but... 70% of energy is lost! (heat, deformation,vibrations,...) Forces are large (>5000 lbs!) Time is short (<1/1000 sec!) The hands don’t matter! Conservation of momentum not full story. Need to say something about inelasticity of collision…after all, a super ball will react differently than a baseball. Need model for ball (static vs. dynamic) Time determined by compressibility (as for a spring) Hands don’t matter: F>>fhands T<<response time of bat

Inelasticity: The Coefficient of Restitution COR = Vrel,f/Vrel,I COR2 = KEcm,f /KEcm,i For baseball, COR=.52-.58 Changing COR by .05 changes V by 7 mph (35 ft!) How to measure? Bounce ball off hard surface COR2 = hf/hi Test procedures: 85 mph on ash/concrete. Not sensitive to “inside”. General principle…deep interior-->higher speed More typical: 160 mph! How does COR depend on relative speed? Dots indicate “allowed range”

What About the Bat? (or, it takes two to tango!) Energy shared between ball and bat Ball is inefficient:  25% returned Wood Bat r~0.02 80% restored COReff = 0.50-0.51 Aluminum Bat r~0.10 COReff = 0.55-0.58 “trampoline effect” ball flies off the bat! r  Ebat/Eball  kball/kbat  xbat/ xball Tennis racket like Al bat. Al bat: possibly 10% in COR==>7 mph==>35’ or more issue: stiffness vs. cor of ball technology of Al bats: thinner wall==>increase r bat: “tennis racket”-like efficient even for dead ball >10% larger!

Properties of Bats length, diameter weight position of center of gravity where does it balance? distribution of weight moment of inertia center of percussion stiffness and elasticity vibrational nodes and frequencies DEMO with real bat. Distribution of weight: 1. How far from handle-->affects swing (NCAA: specifies weight but ignores I) 2. How weight is distributed about CM--->affects energy lost to rotation

Sweet Spot #1: Maximum Energy Transfer Barrel end of bat maximizes bat speed Center of Mass minimizes angular impulse MET must be in between MET  COP @ 5” from knob Aluminum bat more effective for inside pitches CM DISCUSS: Wood vs. Al For Al, more uniform wt. Distribution-->CM closer to handle For same reason, higher I about CM (sort of same reason wider head tennis racket is more effective over a larger region) Both conspire to keep curve higher for inside, lower for outside Actually the shifting of CM is the main effect: would have been more effective for outside pitches if CM were in same place as wood. Alum Wood xcm 21.9” 19.6” kch 9.2” 10.2” kh 23.8” 22.1”

Sweet Spot #2: Center of Percussion When ball strikes bat... Linear recoil conservation of momentum Rotation about center of mass conservation of angular momentum When COP hit The two motions cancel (at conjugate point) No reaction force felt x1 x2 REMARKS: tennis racket golf putter x1x2=Icm/M

Sweet Spot #3: “Node” of Vibration Collision excites bending vibrations in bat Ouch!! Energy lost ==>lower COR Sometimes broken bat Reduced considerably if collision is a node of fundamental mode Fundamental node easy to find For an interesting discussion, see www.physics.usyd.edu.au/~cross DEMO: strike bat (hear and feel): 160 Hz, 560 Hz, …

Dynamics of Bat-Ball Collision Step 1: Solve eigenvalue problem for free vibrations Step 2: Model force Step 3: Expand in normal modes and solve Like solving a nuclear physics problem!

General Results Excitation of normal mode depends on ... fnT (or T/Tn) yn at impact point For T  1 ms only lowest 2 or 3 modes important (fn=171, 568, 1178, 1851,…) General principal of time-frequency

theory vs. experiment (Rod Cross) RESULTS: typical speed theory vs. experiment (Rod Cross) at low speed

Advantages of Aluminum Length and weight “decoupled” Can adjust shell thickness More compressible => “springier” Trampoline effect More of weight closer to hands Easier to swing Less rotational energy transferred to bat More forgiving on inside pitches Stiffer for bending Less energy lost due to vibrations

Aerodynamics of a Baseball Forces on Moving Baseball No Spin Boundary layer separation DRAG! FD=½CDAv2 With Spin Ball deflects wake ==>Magnus force FMRdFD/dv Force in direction front of ball is turning DRAG: Ball has to push air out of the way==>v2 Air follows contours, then breaks off in turbulant wake Result…high pressure in front, low pressure in back-->drag Separation further to front as v increases--> Magnus for 1800 rpm, top/bottom +/- 15 mph! Ball pulls air on top further around than air on bottom.

How Large are the Forces? =1800 RPM Blip around 95 mph: laminar to turbulant boundary layer Drag is comparable to weight Magnus force < 1/4 weight)

The Flight of the Ball: Real Baseball vs. Physics 101 Baseball Role of Drag Role of Spin Atmospheric conditions Temperature Humidity Altitude Air pressure Wind Max @ 350 ROLE OF DRAG: factor of 2 in distance optimum angle from 45 deg to 35 deg not parabolic orbit (outfielders know this! ball goes up, then “dies” and just sort of falls MAGNUS: keeps ball in air longer. Esp impt for golf. 100’ altitude +7’ 10 deg air temp +4’ 10 deg ball temp +4’ 1” drop in barometer +6’ 1 mph following wind +3’ ball at 100% humidity -30’ hit along foul line +11’ approx linear

The Role of Friction Friction induces spin for oblique collisions Spin  Magnus force Results Balls hit to left/right break toward foul line Backspin keeps fly ball in air longer Topspin gives tricky bounces in infield Pop fouls behind the plate curve back toward field

The Home Run Swing The optimum home run angle! Ball arrives on 100 downward trajectory Big Mac swings up at 250 Ball takes off at 350 The optimum home run angle! So how does he do it? Strong==>ability to get high bat speed quickly; good technique Show movie of #70 NOTE: 10o is ideal “contact” angle

Pitching the Baseball “Hitting is timing. Pitching is upsetting timing” ---Warren Spahn vary speeds manipulate air flow orient stitches

Let’s Get Quantitative! How Much Does the Ball Break? 3 4 5 6 7 10 20 30 40 50 60 Vertical Position of Ball (feet) Distance from Pitcher (feet) 90 mph Fastball Kinematics z=vT x=½(F/M)T2 Calibration 90 mph fastball drops 3.5’ due to gravity alone Ball reaches home plate in ~0.45 seconds Half of deflection occurs in last 15’ Drag: v  -8 mph Examples: “Hop” of 90 mph fastball ~4” Break of 75 mph curveball ~14” slower more rpm force larger 0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 Horizontal Deflection of Ball (feet) Distance from Pitcher (feet) 75 mph Curveball Skip; show JHS trajectories instead.

What about split finger fastball? Examples of Pitches Pitch V(MPH)  (RPM) T M/W fastball 85-95 1600 0.46 0.10 slider 75-85 1700 0.51 0.15 curveball 70-80 1900 0.55 0.25 What about split finger fastball?

Obstructions cause turbulance Turbulance reduces drag Effect of the Stitches Obstructions cause turbulance Turbulance reduces drag Dimples on golf ball Stitches on baseball Asymmetric obstructions Knuckleball Two-seam vs. four-seam delivery Scuffball and “juiced” ball skip

Summary Much of baseball can be understood with basic principles of physics Conservation of momentum, angular momentum, energy Dynamics of collisions Excitation of normal modes Trajectories under influence of forces gravity, drag, Magnus,…. There is probably much more that we don’t understand Don’t let either of these interfere with your enjoyment of the game!

What Determines Batted Ball Speed? A Simple Formula Conservation of momentum, energy, and angular momentum: Insect sitting on bat (90+70=160 mph). Violent collision. Ball reverses direction, goes out at 100 mph. Bat exerts large force on ball. Ball exerts equal and opposite force on bat. Bat recoil backwards. Energy of bat is energy robbed from ball. Heavier the bat, less the recoil. Billiard balls… Billiard ball on bowling ball… Billiard ball on brick wall…. Diminishing returns Moreover…speed of bat depends on weight of bat. Tradeoff. Tendency to use lighter bats (30-35 oz): more control; can wait longer; ... NCAA (L-5; L-3) radius of gyration

How Would a Physicist Design a Bat? Wood Bat already optimally designed highly constrained by rules! a marvel of evolution! Aluminum Bat lots of possibilities exist but not much scientific research a great opportunity for ... fame fortune Wood: Not much one can do, given constraints imposed by rules. Aluminum: NCAA!

Example 1: Fastball 85-95 mph 1600 rpm (back) 12 revolutions 0.46 sec M/W~0.1 Gravity: falls ~3.5 ft. Magnus is 0.1 W and up Hence…falls 4” less…enough to cause a problem in a game of inches NOTE: no rising fastball! TOSS STYROFOAM BALL

Example 2: Split-Finger Fastball 85-90 mph 1300 rpm (top) 12 revolutions 0.46 sec M/W~0.1 Falls ~4” more than gravity, 8” more than normal fastball! TOSS STYROFOAM BALL

Example 3: Curveball 70-80 mph 1900 rpm (top and side) 17 revolutions 0.55 sec M/W~0.25 Magnus is larger (spin,vel) Velocity is smaller Total break can be upwards of 18” Usually drops/breaks TOSS STYROFOAM BALL

Example 4: Slider 75-85 mph 1700 rpm (side) 14 revolutions 0.51 sec M/W~0.15 Faster than curveball, less break, side break

Note: both ball and racket compress