1. 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

2. REFERENCES
The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN
The Physics of Sports, Angelo Armenti (American Institute of Physics, New York, 1992), ISBN
L. L. Van Zandt, AJP 60, 72 (1991)
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.

3. “...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:
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 sec in timing, ball moves 1’.
Compare with golf, where ball is not even moving!

4. 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

5. What Determines Batted Ball Speed?
1. pitched ball speed
2. bat speed
Rigid-Body Kinematics:
V = 0.25 Vball 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!

6. 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

7. 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

8. Inelasticity: The Coefficient of Restitution
COR = Vrel,f/Vrel,I COR2 = KEcm,f /KEcm,i
For baseball, COR=
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”

9. 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 =
Aluminum Bat
r~0.10
COReff =
“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!

10. 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

11. 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  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 ” 10.2”
kh 23.8” 22.1”

12. 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

13. 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
DEMO:
strike bat (hear and feel): 160 Hz, 560 Hz, …

14. 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!

15. 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

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

17. 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

18. 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.

19. 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)

20. 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
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

21. 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

22. 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

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

24. 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.

25. What about split finger fastball?
Examples of Pitches
Pitch V(MPH)  (RPM) T M/W
fastball
slider
curveball
What about split finger fastball?

26. 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

27. 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!

28. 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

29. 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!

30. 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

31. 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

32. 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

33. 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

34. Note: both ball and racket compress