Hitting Home Runs:
How a Physicist Thinks About Baseball
Alan M. Nathan
University of Illinois at Urbana-Champaign a-nathan@uiuc.edu
http://www.npl.uiuc.edu/~a-nathan/pob

My day job…
 experimental nuclear/particle physics
 high-speed collisions between subatomic particles
 Nights and weekends...
 physics of baseball
 high-speed collision between baseball and bat
Jefferson Laboratory Science Series March 28, 2002 Page 1

“...the most difficult thing in sports”
--Ted Williams
BA: .344
SA: .634
HR: 521
OBP: .483
#521, last at-bat
September 28, 1960
Jefferson Laboratory Science Series March 28, 2002 Page 2



90 mph fastball takes about 0.40 sec to reach batter
 ~ 0.20 sec needed to “observe, process, decide”
 ~ 0.15 sec needed for swing
 half of “break” occurs in last 0.10 sec if batter mis-estimates speed by

1.5 mph (0.007 sec)
 swing will be early/late by 1’
 elevation of ball misjudged by
 1”
 backspin/topspin makes ball fall less/more
 Warren Spahn (greatest lefthander?) said:
 “Hitting is timing. Pitching is upsetting timing.”
 30% success rate gets you in Hall of Fame!
Jefferson Laboratory Science Series March 28, 2002 Page 3

 A violent collision!
 forces large (>8000 lbs!)
 time is short (<1/1000 sec!)
 ball compresses, stops, expands
 kinetic energy
 potential energy
 lots of energy dissipated
 distortion of ball and bat
 hands don’t matter!
Jefferson Laboratory Science Series March 28, 2002 Page 4

 Large hit ball speed
 v hit

105 mph

D

400 ft
 each additional mph gives 4-5 ft
 Proper takeoff angle
 about 35 0
 Lots of backspin
Jefferson Laboratory Science Series March 28, 2002 Page 5


The basic stuff (“kinematics”)
 speed of pitched ball
 speed of bat
 weight and weight distribution of bat

The really interesting stuff (“dynamics”)
 “bounciness” of ball
 impact location on bat (“sweet spot”)
Jefferson Laboratory Science Series March 28, 2002 Page 6

 My only formula v hit
ball
bat example: 90 + 70 gives 102 mph (~400”)
 Bat speed matters much more!
 Each mph of bat speed worth 5 ft
 Each mph of pitch speed worth 1 ft
 Collision is very inefficient
 perfect efficiency: 0.2/1.2

1.0/2.0
Jefferson Laboratory Science Series March 28, 2002 Page 7

v hit
= 0.2 v ball
+ 1.2 v bat
CM recoil rotation
• For given bat speed, heavier bat is better
0.2/1.2

0.5/1.5
• But….
Jefferson Laboratory Science Series March 28, 2002 Page 8


COR measures “bounciness” of ball
 Final speed/Initial speed
(from massive rigid surface)
 For baseball, COR

0.5
 h f
/h i
= 3/4
 3/4 energy lost!

Is the ball “juiced ”?
 10% change
 30’
Jefferson Laboratory Science Series March 28, 2002 Page 9

 Collision excites bending vibrations in bat
 Ouch!! Thud!!
 Sometimes broken bat
 Energy lost
 lower COR
Jefferson Laboratory Science Series March 28, 2002 Page 10

nodes Center of mass
“sweet spot” zone
Jefferson Laboratory Science Series March 28, 2002 Page 11

 thin shell & added mass
 mass, mass distribution, length independent
 lighter and CG closer to hands
 …easier to swing
 fatter barrel, thinner handle
 more forgiving on mis-hits
 “trampoline” effect
 golf drivers
 tennis rackets
 corked bats?
 Net result: higher batted ball speed
Jefferson Laboratory Science Series March 28, 2002 Page 12

Forces on Moving Baseball
Without Spin:
 Ball pushes air out of way
 Low-pressure wake

DRAG!
 Grows with square of velocity
With Spin:
 Ball deflects wake
 action/reaction
 “Magnus force”
 Force grows with rpm
 Force in direction front of ball is turning
Jefferson Laboratory Science Series March 28, 2002 Page 13

The Flight of the Ball
Real Baseball vs. Physics 101 Baseball
Jefferson Laboratory Science Series March 28, 2002 Page 14

Friction between ball and bat
 spin on ball
Bat undercuts ball

• ball goes up
• backspin
Jefferson Laboratory Science Series March 28, 2002 Page 15

Friction between ball and bat
 spin on ball
Bat overcuts ball

• ball goes down
• topspin
Jefferson Laboratory Science Series March 28, 2002 Page 16

Friction between ball and bat
 spin on ball
Swing early/late

• ball goes left/right
• sidespin
Jefferson Laboratory Science Series March 28, 2002 Page 17

 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
Jefferson Laboratory Science Series March 28, 2002 Page 18

Backspin (RPM) optimum angle distance
0
500
36 0
35 0
355 ft
361 ft
1000
1500
2000
2500
34 0
32 0
31 0
29 0
368 ft
375 ft
382 ft
389 ft
Jefferson Laboratory Science Series March 28, 2002 Page 19

• Ball arrives on 8.5
o downward trajectory
• Big Mac swings up at 8.5
o
• Undercuts ball by ~1”
• Ball takes off at 35 o with lots of backspin
The optimum home run swing!
Jefferson Laboratory Science Series March 28, 2002 Page 20

 Lots of physics in baseball
 Knowledge of physics helps understanding of game

And it’s fun too….
Jefferson Laboratory Science Series March 28, 2002 Page 21