1. The Physics of Baseball (or…Just How Did McGwire Hit 70. ) Alan M
The Physics of Baseball (or…Just How Did McGwire Hit 70?) Alan M. Nathan University of Illinois February 5, 1999
Introduction
Hitting the Baseball
The Flight of the Baseball
Pitching the Baseball
Summary
Personal note: physics all my professional life; baseball much longer; connection only recently.
But you also know that you really don’t have to be Einstein to understand or play the game. After all, it isn’t rocket science, or brain surgery, or even nuclear physics. But there is more to the game than meets the untrained eye. In my day job as an nuclear physicist, I do experiments involving the collision of subatomic particles. It really is no exaggeration that some of the same physics principles that apply in those collisions also applies in a very different type of collision, that between the baseball and bat. So, in this talk, I hope to give you an appreciation of some of the ways that physics can help describe the game of baseball and in the process give you a whole different way of looking at the game. It won’t make you another Mark McGwire but it should take you up a notch in your level of appreciation.
So, with that in mind, here is an outline of my talk.
Now, let’s play hardball! 1

2. REFERENCES
The Physics of Baseball, Robert K. Adair (Harper Collins, New York, 1990), ISBN
The Sporting Life, Davis and Stephens (Henry Holt and Company, New York, 1997), ISBN
ME!
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.
Goal is not to use physics to improve the game. Through over 100 years of evolution, ball players know how to play the game, even though they may not really know why they do what they do. They know what works. Our goal is to understand the game.
So…let’s begin

3. “...the most difficult thing to do in sports”
Hitting the Baseball
“...the most difficult thing to do in sports”
--Ted Williams,
Professor Emeritus of
Hitting
Round ball, round bat, squarely
<0.5 s; half of break after decision reached; timing (.01s -->1’)
Physiological problem…energy from muscles to bat
Lots of ways to fail (early, late, up, down, along the bat…)
Compare with golf, where ball is not even moving!
Hard!
Compare with subatomic collisions--fascinating

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
What features of collision maximize speed of hit ball as it leaves the bat?
NOTE: for home runs….105 mph/400’, 1mph/5’
GO TO JHS DEMO
80,60-->102 90,60-->105 90,70-->118

5. What Determines Batted Ball Speed?
How does batted ball speed depend on ...
pitched ball speed?
bat speed?
V = 0.25 Vball Vbat
1. SHOW GRAPHS: Bat speed more important than ball speed, although both are relevant. You can hit home from lobbed pitch (or fungo), perhaps with some difficulty, but you surely cannot hit homer by bunting. This agrees with your intuition. How can we understand the physics of this?
Formula: 10 mph of ball speed is worth 2.5 mph
10 mph of bat speed is worth 12.5 mph
2. DISCUSS: Principle is conservation of momentum. Momentum is mass times velocity. Bat has much more of it than ball because it is so much heavier. During the collision, some of the bats momentum is transferred to the ball, turning it around and sending it out with a larger speed than it had coming in. The bat loses an equal amount of momentum, but the total momentum of bat plus ball is the same before as after the collision. It is because the bat is so much heavier than the ball that it’s speed is much more important than the pitched ball speed. Our Physics 111 students actually do this situation as a homework problem, so the physics is fairly basic.
Conclusion:
Bat Speed Matters More!

6. What Determines Batted Ball Speed?
Mass of bat
Conclusion:
mass of bat matters
...but not a lot
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)

7. Dynamics of Ball-Bat Collision
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!
SO FAR: mass,speed of ball,bat; conservation of momentum;
before, after collision
NOT ENOUGH: superball
NEED TO LOOK MICROSCOPICALLY (DURING)
The interesting stuff: where does energy go? How does this affect what happens? What are implications for bat/ball design?
Nuclear physics analogy!
Important point: time is very short. Ball doesn’t know the hands are there. You could let go of the bat 1 msec before collision and it would make no difference! (a controversial statement)

8. Dynamics of Ball-Bat Collision
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!
SO FAR: mass,speed of ball,bat; conservation of momentum;
before, after collision
NOT ENOUGH: superball,nurf ball (SHOW THESE)
NEED TO LOOK MICROSCOPICALLY (DURING)
The interesting stuff: where does energy go? How does this affect what happens? What are implications for bat/ball design?
Nuclear physics analogy!
Important point: time is very short. Ball doesn’t know the hands are there. You could let go of the bat 1 msec before collision and it would make no difference! (a controversial statement)

9. The Coefficient of Restitution
COR measures “bounciness” of ball
Final speed/Initial speed
For baseball, COR=
Changing COR by .05 changes V by 7 mph (35 ft!)
How to measure?
This is square of COR >
Conservation of momentum not full story. Try hitting superball!
All other things being equal, superball will go much further!
Bounciness, or elasticity, of ball. Has to do with amount of kinetic energy--energy of motion--lost in the collision.
DEMO: DROP BASEBALL FROM FLOOR; DROP SUPERBALL FROM FLOOR.
Whatever this bounciness is, superball has more of it.
MODEL (the way physicists work): springs, friction, lost energy.
COR. Same model tells us size of forces and time over which they act (has to do with vibrational motion of springs).
Test procedures: 85 mph on ash/concrete. Not sensitive to
“inside”. General principle…deep interior-->higher speed
More typical: 160 mph!

10. What About the Bat? (or, it takes two to tango!)
Wood Bat
Efficiently restores energy
But only 2% energy stored
Bat Performance Factor (BPF) ~1 .02
Aluminum Bat
Stores ~ 20% energy
Result: “trampoline effect”
BPF ~ 1.2
Ball flies off the bat!
A more efficient bat and/or ball
DEMO: BASEBALL ON FLOOR, TENNIS RACKET
DEAD BALL ON FLOOR, TENNIS RACKET
SUPERBALL ON FLOOR, SPONGE
For well-hit ball (near sweet spot)...
The issue: springiness of surface. Bat compresses ball, ball compresses bat--same size force--which one gives depends on how easily compressed. Wood bat, for given force, compresses 1/50 of ball. Or…only 1/50 of energy in collision is stored in bat, rest is in ball.
Tennis racket--effective BPF>1, just like Al bat.
Al bat: 10% in COR==>7 mph==>35’ or more
COST!
new rule changes in NCAA
nails!
Better bat: a tennis racket
Better ball: spherical metallic shell

11. 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
Stiffness affects energy lost to vibrations in bat

12. Sweet Spot #1: Center of Percussion
When ball strikes bat...
Linear recoil
conservation of momentum
Rotation about center of mass
conservation of angular momentum
When CP hit
The two motions cancel at handle
No reaction force felt at handle
CM: define
Hit at CM: recoils (momentum conservation)
Hit anywhere else: recoils and rotates (two separate motions)
Angular Momentum conversation:
Amount of recoil independent of where you hit it.
Amount of rotation get larger the further from CM hi whack it
DEMO with swinging bat.
REMARKS:
tennis racket
golf putter

13. Sweet Spot #2: Maximum Energy Transfer
Barrel end of bat maximizes bat speed
Center of Mass minimizes angular impulse
MET must be in between
Not on COP!
Aluminum bat more effective
for inside pitches CM
COP
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

14. 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:
wiggle nurf bat
strike bat (hear and feel): 160 Hz, 560 Hz, …
NOTE: nuclear physics analogy
Principle: time/frequency
Al bat: stiff to bending, esp in handle region--> higher frequencies (200 Hz vs 165 Hz)-->less effectively excited in collision. More forgiving for inside pitch!
NOTE: no hands approx probably breaks down for balls hit far from node. I.e., firm grip helps for mishit balls.

15. So you think bats cannot bend…..

16. So you think bats cannot bend…..

17. 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!

18. 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

19. Aerodynamics of a Baseball
Forces on Moving Baseball
No Spin
Boundary layer separation
DRAG!
Grows with v2
With Spin
Ball deflects wake
action/reaction==>Magnus force
Force grows with rpm
Force in direction front of ball is turning
DRAG:
Ball has to push air out of the way
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
Ball pulls air on top further around than air on bottom.

20. The Flight of the Balll Role of Drag Role of Spin
Atmospheric conditions
Temperature
Humidity
Altitude
Air pressure
Wind
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’

21. 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. 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

24. Pitching the Baseball Don Larsen, 1956 World Series
Last pitch of perfect game
“Hitting is timing. Pitching is
upsetting timing”
---Warren Spahn
vary speeds
manipulate air flow
orient stitches

25. Let’s Get Quantitative! I. How Large are the Forces?
Terminal velocity
Drag is comparable to weight
Magnus force < 1/4 weight)

26. Let’s Get Quantitative! II. How Much Does the Ball Break?
Depends on…
Magnitude and direction of force
Time over which force acts
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 reduces fastball by about 8 mph
Examples:
Hop of 90 mph fastball: ~4”
Break of 70 mph curveball ~16”
slower
force larger
Skip; show JHS trajectories instead.

27. 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

28. 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

29. 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

30. 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

31. Examples of Trajectories3 4 5 6 7 10 20 30 40 50 60
Vertical Position of Ball (feet)
Distance from Pitcher (feet)
90 mph Fastball
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

32. 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

33. Summary Much of baseball can be understood with
basic principles of physics
Conservation of momentum, angular momentum, energy
Dynamics of collisions
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!