1. Analysis of Knuckleball Trajectories
Alan M. Nathan
University of Illinois at Urbana-Champaign
Trampoline effect “universal”: golf, tennis, baseball/softball, etc.. Physics is the same in each.
Recently retired knuckleball pitcher Tim Wakefield 1

2. Issues to be Addressed The “movement” of knuckleball pitches
The “smoothness” of knuckleball trajectories

3. Knuckleball thrown with very little spin  no Magnus force
But still lots of erratic “movement”
Origin of movement revealed in wind tunnel experiments

4. Wind Tunnel Data, 4S Orientation
Mike Morrissey (MS Thesis) and John Borg (Marquette)
Agrees with Watts & Sawyer, AJP (1975)

5. Studying Knuckleball Trajectories Using the PITCHf/x Tracking System
Two video fps
approximately orthogonal axes
full 3D reconstruction
tracks every pitch in every MLB ballpark
all data publicly available
Image, courtesy of Sportvision

6. Studies of Knuckleball Movement
Movement = deviation of trajectory from straight line, with gravity removed
Easily measured with PITCHf/x
View from above
5” movement

7. Direction of movement vs. release speed
(Jon Lester)
Catcher’s View
“Normal” pitches have predictable movement

8. Direction of movement vs. release speed
(Tim Wakefield)
Knuckleballs do not have predictable movement

9. But is the trajectory “smooth”?
Fit to smooth function
Examine RMS deviation of data from fit
9 Free Parameters:
x0, y0, z0, vx0, vy0, vz0, CD, CL, 

10. 278 pitches from August 29, 2011 Knuckleball (77) Normal (201)
Normal and knuckleball pitches follow similar distributions
Knuckleballs only slightly (few tenths of inch) less smooth

11. Two Examples: Which one is the knuckleball?
76 mph knuckleball
rms=0.374”
75 mph curveball
rms=0.373”

12. 2011Aug

13. Summary of Conclusions
Movement of knuckleball trajectories varies considerably from pitch to pitch
Magnitude and direction quasi-random
Any given trajectory is as smooth as those of ordinary pitches
within limits of precision of tracking data (~0.3”-0.5”)
Open questions/work in progress
Are erratic movement and smoothness conclusions consistent with wind tunnel data?
How can perception and reality be reconciled?