1. Physics and Baseball: A Report to Red Sox Nation
Alan Nathan, University of Illinois

2. A good book to read…. Skip Prof. Bob Adair
“…the physics of baseball is not the clean, well-defined physics of fundamental matters. Hence conclusions must depend on approximations and estimates. But estimates are part of the physicist’s repertoire...”
“The physicist’s model of the game must fit the game.”
“Baseball is not rocket science. It’s much harder.”
Skip

3. Greatest baseball team
The Baseball/Physics Connection
1927 Yankees:
Greatest baseball team
ever assembled
MVP’s
1927
Solvay Conference:
Greatest physics team
ever assembled

4. Topics I Will Cover The ball-bat collision The flight of the baseball
How a bat works
Wood vs. aluminum
The flight of the baseball
Drag, lift, and all that
New tools for baseball analysis

5. “You can observe a lot by watching” ---Yogi Berra
forces large, time short
>8000 lbs, <1 ms
ball compresses, stops, expands
like a spring: KEPEKE
bat recoils
lots of energy dissipated
distortion of ball
vibrations in bat
Hold ball and bat
Ball compresses to half its radius (nearly an inch!)
Takes a lot of force to do that
Well hit ball has large BBS
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

6. What Determines Batted Ball Speed?
pitch speed
bat speed
“collision efficiency”: a property of the ball and bat
BBS = q vpitch + (1+q) vbat
typical numbers: q = q = 1.2
example: gives 101 mph (~400’)
vbat matters much more than vpitch!
Each mph of bat speed worth ~6 ft
Each mph of pitch speed worth ~1 ft
Superball on massive wall: e~1.

7. Kinematics of Ball-Bat Collision
BBS = q vpitch + (1+q) vbat
1. m/Meff = ball mass/effective bat mass  0.25
bat recoil
2. e = elasticity of collision  0.50
energy dissipation
For m/Meff <<1 and e1, q1
Superball on massive wall: e~1.

8. 1. Effective Bat Mass
Meff  “Swing Weight”: related to MOI about the handle
Larger  less recoil to bat  larger q
Larger  smaller swing speed
Batters seem to prefer lower MOI bats sacrificing power for “quickness”
Can transfer more energy to ball with heavier than lighter object—or, less recoil to bat.
Ideal bat wt: Depends critically on how bat speed depends on bat weight.
Not really a physics issue but a biomechanics issue.
Power vs. control
Cross and AMN, Sports Technology 2, 7-15 (2009)

9. Is There an Advantage to “Corking” a Bat?
Sammy Sosa, June 2003
Gain in bat speed offset by reduction in collision efficiency
Based on best experimental data available:
…for “harder” hit: no
…for frequency of good contact: probably

10. e = ball-bat coefficient of restitution (bbcor)
1 - e2 = fraction of CM energy dissipated
~75%!
Joint property of ball and bat
Most of energy loss is in ball
But the bat matters
Vibrations decrease e
Trampoline effect increase e
Bounce some balls:
Golf ball
Steel ball
Baseball
Happy ball
Sad ball

11. Vibrations and the ball-bat collision
Demo with…
Tapping bat to listen for node
outside
“sweet spot”

12. Studying the Vibrations of a Baseball Bat
f1 = 179 Hz
f2 = 582 Hz
f3 = 1181 Hz
f4 = 1830 Hz
time
frequency
Note: nodes stack up at barrel end-->”sweet spot zone”

13. COR Dynamics of the Bat-Ball Collision y z
AMN, AJP 68, (2000)
F=kxn 20
Solve eigenvalue problem for normal modes
Model ball-bat force F
Expand y in normal modes
Solve coupled equations of motion for ball, bat
Energy budget:
KE of ball (batted ball speed)
recoil of bat
dissipation in ball
vibrations in bat
Two “free” parameters…Young’s and shear modulus
COR

14. Vibrations, BBCOR, and the “Sweet Spot”
at ~ node 2
vibrations minimized
COR maximized
BBS maximized
best “feel” + e vf
COR vs impact location
E-vib vs impact location
Vf vs impact location
Response in handle vs impact location
Sweet spot (feel): close to node 2 (hands insensitive to higher modes; hands are already at node 1)
Evib

15. Independence of End Conditions
strike bat on barrel—look at movement in handle
handle moves only after ~0.6 ms delay
collision nearly over by then
nothing on knob end matters
size, shape, hands, grip
boundary conditions
confirmed experimentally
Batter could drop bat just before contact and it would have no effect on ball!!!

16. BBCOR and the Trampoline Effect (hollow bats)
The Ping!
Lowest Hoop (or wineglass) Mode

17. The “Trampoline” Effect: A Simple Physical Picture
BBCOR increases with …
elasticity of ball (~0.5)
elasticity of bat (~1)
relative stiffness ~ kball/kbat
BBCOR(Al)/BBCOR(wood)
unregulated, can be very large
Little League <1.15
NCAA < 1.0 (!)
wood
alum
Change kball
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?
change kbat

18. Forces on a Spinning Baseball in FlightFM
Drag slows ball down
Magnus + mg deflects ball from straight line FD mg
Won’t discuss CM here except to say that data on golf balls suitably scaled for a baseball indicate that time constant for spin decay is large, in the 20’s of seconds or more, so that it is safe to ignore, at least until we get better data on it.

19. Real vs. “Physics 101” Trajectory: Effect of Drag and Magnus
Drag reduces distance by a lot. Make trajectory asymmetric. Reduces optimum angle
Magnus gives lift, keeps ball in air longer. Reduces optimum angle even further.

20. What do we know about CD? (mainly from pitch tracking)
Depends on ….
v0 (Reynold’s Number)
surface “roughness”?
seam orientation?
spin?
Dedicated TrackMan
Oct. 2008
StL, Sept. 2009
PITCHf/x
TrackMan
Skip
Good approximation:
Cd = 0.35±0.05 in range mph
No steep “drag crisis”
More dedicated experiments in progress

21. What do we know about CL? (mainly from high-speed motion analysis)
Depends on ….
spin parameter S  R/v
fixed S?
best evidence is “no”, in region of mph
seam orientation?
Skip
Good approximation:
CL S  R/v in range

22. New tools to study flight of baseball
PITCHf/x and HITf/x
Video tracking
TrackMan
Doppler radar tracking

23. PITCHf/x and HITf/x Two video cameras @60 fps
Marv White, Physics,
UIUC, 1969
Marv White, Physics,
UIUC, 1969
Two video fps
“high home” and “high first”
tracks every pitch in every MLB ballpark
all data publicly available on web!
tracks initial trajectory of batted ball
Used for analysis, TV broadcasts, MLB Gameday, etc.
Image, courtesy of Sportvision

24. TrackMan Skip Doppler radar to measure radial velocity
dr/dt  r(t)
3-detector array to measure phase
two angles (t), (t)
Together these give full 3D trajectory
Spin modulates to give sidebands
spin frequency 
Skip

25. So what good is a physicist in all this?
Minimal parametrization of the trajectory
Constant acceleration works very well for pitched ball
Constant “jerk” works for most batted balls
Keeping everyone honest
Laws of physics cannot be violated
Recognizing and dealing with imperfect data
Measurements have uncertainties!
Skip

26. Baseball Analysis: Using PITCHf/x to discover how pitchers do what they do
“Hitting is timing. Pitching is upsetting timing.”

27. Location, Location, Location
Ex 1: Mariano Rivera: Why is he so good??
home plate
Home Runs
Three Reasons:
Location, Location, Location

28. Ex 2: “Late Break”: Truth or Myth Mariano Rivera’s Cut Fastball
View from above:
actual trajectory
linear extrapolation
Both pitches heading for ~same location. One seems to break at the last minute, the other doesn’t.

29. Ex 2a: What makes an effective slider
Josh Kalk, THT, 5/22/08
This slider is very effective since it looks like a fastball for over half the trajectory, then seems to drop at the last minute (“late break”).
side view

30. Ex 3: A Pitcher’s Repertoire
4-seam fastball
slider/cutter
2-seam fastball
changeup
curveball
Remove spin axis arrows. Try to find ball with spin axis showing.
Catcher’s View

31. Ex 4 Jon Lester vs. Brandon Webb
15 inches
Remove spin axis arrows. Try to find ball with spin axis showing.
Brandon Webb is a “sinkerball” pitcher:
Almost no rise on his fastball

32. Tim Wakefield is a knuckleball pitcher:
Ex 5 The Knuckleball
Remove spin axis arrows. Try to find ball with spin axis showing.
Tim Wakefield is a knuckleball pitcher:
Chaotic Movement

33. Ex 5 Ubaldo Jimenez Pitching at High Altitude
San Diego
Denver
vf/v0
vf/v0
"Every time that I come here to San Diego, it's always good. Everything moves different. The breaking ball is really nasty, and my fastball moves a lot. So I love it here."
Remove spin axis arrows. Try to find ball with spin axis showing.
Denver
Denver
Denver
Denver

34. Learning About Batted Balls
HITf/x
Initial part of trajectory
All April 2009 data available
TrackMan
Full trajectory
Limited data from StL, Sept. 2009

35. TrackMan Data from StL, 2009 R vs. v0 R vs. 0 USEFUL BENCHMARK
mph
~5 ft per mph
25o-35o

36. What Constitutes a Well-Hit Ball?
w/o home runs HR
BABIP
home runs
V0>90

37. Putting Spin on Batted Balls
in front or behind  sidespin
sideways Magnus force
fly balls break toward foul pole
friction
normal force

38. undercutting/overcutting  backspin/topspin
Magnus force is up/down
Topspin makes line drives nose-dive
Backspin keeps fly ball in air longer
Tricky popups to infield
friction
normal force v
???

39. Paradoxical Popups AJP 76, 723-729 (2008)

40. Combining HITf/x with Hittracker
HITf/x  v0,,
Hittracker (Greg Rybarczyk, hittrackeronline.com)
Landing point
Flight time
Together these constrain the full trajectory

42. HITf/x + hittracker Analysis: The “carry” of a fly ball
(379,20,5.2)
Motivation: does the ball carry especially well in the new Yankee Stadium?
“carry” ≡ (actual distance)/(vacuum distance)
for same initial conditions

43. HITf/x + hittracker Analysis: 4354 HR from 2009
Denver
Cleveland
Yankee Stadium

44. Work in Progress
Collision experiments & calculations to elucidate trampoline effect
New studies of drag and Magnus
Experiments on high-speed oblique collisions
To quantify spin on batted ball

45. 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
go.illinois.edu/physicsofbaseball
I am living proof that knowing the physics doesn’t help you play the game better!
@ Red Sox Fantasy Camp, Feb. 1-7, 2009

46. HITf/x + hittracker Analysis: 4354 HR from 2009

47. CD: One Final Thought PFX TM PFX-TM
Correlations suggestive of variations in baseball

48. Extract sidespin vs.  from trajectoryCF LF RF RF LF RF
LHH
break to right
break to left
RHH 
Balls break toward foul pole
Break increases with angle
Ball hit to CF slices
LHH/RHH asymmetry
Tilt in bat

49. Is the Baseball “Juiced”? Is COR larger than it used to be?
Measurements with high-speed cannon
COR=rebound speed/initial speed
1975 vs. 2004
Mention v dependence of COR….better to measure at 150 mph than at 60 mph.
1975 and 2004 equal to few %
No evidence for juiced ball

50. Example: Pitching at High Altitude
10%
loss of velocity
total movement
12”
7.5% 8”
Denver
Toronto
Toronto
Denver
PITCHf/x data contain a wealth of information about drag and lift!