Studies of Batted Ball Trajectories

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Studies of Batted Ball Trajectories
Alan M. Nathan University of Illinois

• I. Analyzing the FFX trajectories
• II. Determining landing point/hang time from HFX
• Combining HFX and Hittracker
• Do drag coefficients vary with ball?

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I. Analyzing FFX Trajectories

• WWAD What Would Alan Do?
• Actually, what DID Alan do?
• Scottsdale, March 2009 experiment
• 10 Cameras uses
• 2 PFX/HFX cameras
• 8 IP cameras
• All data used to analyze trajectories
• PFXHFXFFX

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Analyzing FFX Trajectories

• Track pitch9P PFX
• Track initial batted ball6P HFX
• Get intersection of batted ball and pitched ball
  trajectories to establish contact time
• Track batted ball using FFX cameras
• Do constant acceleration fit to first 0.5 sec of
  FFX data
• Key step Velocity vector fixed at HFX value
• Look for intersection with HFX trajectory to
  synchronize IP and HFX clocks
• Now fit the synchronized FFX and HFX data to
  using your favorite model

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Analyzing FFX Trajectories

• Modeling the batted ball trajectories
• Piecewise (0.5 sec) constant acceleration
• Constant jerk (12P) might work
• Nonlinear model with drag, Magnus, wind, will
  work best
• Possible compromises
• 9Por 10P models Initial position and velocity
  vectors (6) plus constant Cd (1) and spin vector
  (2 or 3)

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Examples Using 9P and 12P

• 12P constant jerk
• Initial positions, velocities, accelerations
• Rate of change of acceleration (jerk)
• 9P aerodynamic model
• Initial positions, velocities
• Constant drag coefficient
• Backspin and sidespin
• Both models utilize nonlinear L-M fitting applied
  to pixels directly

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Fly Ball
V096 mph ?016 deg
Line Drive
Line Drive
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Topspin Line Drive
V0106 mph ?06 deg
Fly Ball
Line Drive
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Incomplete Long Fly Ball
V0104 mph ?023 deg
Fly Ball
Line Drive
9
Line Drive
V099 mph ?07 deg
Fly Ball
Line Drive
Line Drive
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Bad Fit
V0101 mph ?06 deg
Fly Ball
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Some Remarks

• 12P and 9P work equally welI
• Sometimes bad fits
• Probably bad fits due to bad data, not bad model
• 12P provides handy way to parametrize the
  trajectory
• The Arizona data came from an initial experiment.
  Quite possibly the current setup in SF provides
  higher quality data
• I recommend further studies of this type
• Side note the FFX data can be used to correct
  the HFX data, which systematically underestimates
  v0 and ?0

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II. Determining landing point/hang time from HFX

• Utilize ball tracking data from 2009, 2010
• 2900 batted balls
• 2367 batted balls with VLAgt0
• Initial velocity (BBS, VLA, Spray angle)
• Location when z0 and hang time (extrapolated)
• Not a theoretical analysis based entirely on
  data

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Total Distance
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Fit vs Data Distance RMS25 ft Hang
Time RMS0.4 sec Bearing RMS8 deg
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Summary

• Distance RMS25 ft
• Hang Time RMS0.4 sec
• Bearing RMS8 deg
• (Data precision almost surely more accurate
• It is hard to do any better than this without
  additional information (spin? wind? )
• Is it good enough?
• What about reverse (Hittracker)?

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III. Combining HFX with Hittracker

• HITf/x ? (v0,?,?)
• Hittracker ? (xf,yf,zf,T)
• Together ? full trajectory
• HFXHTT determine unique Cd, ?b, ?s
• Full trajectory numerically computed (9P)
• T ? ?b
• horizontal distance and T ? Cd
• sideways deflection ? ?s
• Analysis for gt8k HR in 2009-10

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How well does this work?

• Test experimentally using radar tracking device

• For this example it works amazingly well!
• A more systematic study is in progress

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Ex. 1 The carry of a fly ball

• Motivation does the ball carry especially well
  in the new Yankee Stadium?
• carry (actual distance)/(vacuum distance)
• for same initial conditions

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HITf/x Hittracker Analysis4354 HR from 2009
Denver
Cleveland
Yankee Stadium
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Ex. 2 Effect of Air Density on Home Run Distance
20092010 HR
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The Coors Effect
26 ft
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Phoenix vs. SF
Phoenix 5.5 ft
SF -5.5 ft
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Ex. 3Whats the deal with the humidor?

• Coors Field in Denver
• Pre-humidor (1995-2001) 3.20 HR/game
• Post-humidor (2002-1020) 2.39 HR/game
• 25 reduction
• Can we account for reduction?
• How does elevated humidity affect ball COR and
  batted ball speed?
• How does reduced batted ball speed affect HR
  production?
• See Am J Phys, June 2011

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HR Humidors The Method

• Measure ball COR(RH)
• From 30 to 50, COR decrease by 3.7
• Measurements _at_ WSU (Lloyd Smith)
• Physics ball-bat collision model
• Batted ball speed (BBS) reduced by 2.8 mph
• HittrackerHITf/x
• We know landing point, distance/height of nearest
  fence
• Calculated new trajectory with reduced BBS
• Mean HR distance reduced by 13 ft
• Does ball make it over the fence?

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HR Humidors Results

• The result
• 27.0 ? 4.3 calculated
• 25 actual (!)
• Side issue
• If humidor employed in Phoenix, predicted
  reduction is 37.0 ? 6.5

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Ex. 4 And what about those BBCOR bats?

• Starting in 2011, NCAA regulates non-wood bats
  using bbcor standard
• BBCORball-bat coefficient of restitution
• For wood, ?0.498
• For nonwood, gt0.500 due to trampoline effect
• New regulations bbcor?0.500

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BBCOR bats The Method

• Physicsball-bat collision model
• 5 reduction in BBS
• Hittracker HFX
• Reduction in fly ball distance
• Reduction in HR

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Normalized HR vs. Reduction in BBS
60 reduction
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NCAA Trends in Home Runs
Actual Reduction 50 science works!
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Additional Comments

• This technique can be used to investigate many
  different things such as
• Effect of changing the COR of the baseball
• Effect of moving or changing height of fences
• Implications of a higher swing speed

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IV. Does Cd Vary with Ball?
PFX
TM
0.033
0.032
PFXTM
PFX-TM
0.023
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• Data suggest some measurement-independent
  variation in Cd
• RMS from measurement 0.016
• RMS in common 0.028
• Is the common due to variations in the ball?

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• Analysis
• Find grand average of Cd over all pitches
• Identify consecutive pitches with same ball
• Get mean Cd for each ball i
• Shift Cd for each pitch so that ball
  averagegrand average
• Compare with original distribution of Cd
• Perform same procedure on random pitches
• Analysis uses 22k pitches
• 3.7k involve at least three pitches with same
  ball
• 1.1k different balls
• 0.96k in 90-92 mph range

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Adjusted
Raw
RANDOM
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Conclusions About Cd

• There is compelling evidence that Cd varies
  significantly with ball
• Perhaps as much as 8 RMS
• Measurement variation is less
• A controlled experiment is planned
• Is this information useful to anyone?
• (e.g., Rawlings)

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In Conclusion

• Thanks to all those who provided me with data
• Thanks to Rand Pendelton for lots of interesting
  discussions
• Thanks to all of you for patiently listening
• And now that you think you understand everything,
  have a look at this
• Garcia video removed to save space