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### The Physics of Hitting a Home Run

Thanks to J. J. Crisco & R. M. Greenwald

Medicine & Science in Sports & Exercise 34(10): 1675-1684; Oct 2002

Alan M. Nathan,University of Illinois

www.npl.uiuc.edu/~a-nathan/pob

a-nathan @uiuc.edu

UBC Colloquium 10/5/06

Greatest baseball team

ever assembled

1927

Solvay Conference:

Greatest physics team

ever assembled

MVP’s

Baseball and Physics

UBC Colloquium 10/5/06

Adair’s Book: An Excellent Reference

“Our goal is not to reform the game but to understand it.

“The physicist’s model of the game must fit the game.”

UBC Colloquium 10/5/06

The Physics of Hitting a Home Run

- How does a baseball bat work?
- Aerodynamics: flight of a baseball
- Leaving the no-spin zone
- Putting it all together

UBC Colloquium 10/5/06

“You can observe a lot by watching”

Champaign News-Gazette

--Yogi Berra

Easton Sports

CEComposites

UBC Colloquium 10/5/06

Brief Description of Ball-Bat Collision

- forces large, time short
- >8000 lbs, <1 ms
- ball compresses, stops, expands
- KEPEKE
- bat recoils
- lots of energy dissipated (“COR”)
- distortion of ball
- vibrations in bat
- to hit home run….
- large hit ball speed (100 mph~400 ft)
- optimum take-off angle (300-350)
- lots of backspin

UBC Colloquium 10/5/06

vbat

vf

Kinematics of Ball-Bat Collision

vf = q vball + (1+q) vbat

- q “Collision Efficiency”
- Joint property of ball & bat
- independent of reference frame
- ~independent of “end conditions”—more later
- weakly dependent on vrel
- Superball-wall: q 1
- Ball-Bat near “sweet spot”: q 0.2
- vf 0.2 vball + 1.2 vbat

Conclusion:

vbat matters much more than vball

UBC Colloquium 10/5/06

vbat

vf

q=0.20

Kinematics of Ball-Bat Collision

- r = mball /Mbat,eff :bat recoil factor = 0.25
- (momentum and angular momentum conservation)
- ---heavier is better but…
- e:“coefficient of restitution” 0.50
- (energy dissipation—mainly in ball, some in bat)

UBC Colloquium 10/5/06

Collision Efficiency q Can Be Measured

- Air cannon to fire ball onto stationary bat
- q = vout/vin
- Used by NCAA, ASA, … to regulate/limit performance of bats

Sports Sciences Lab @ WSU

UBC Colloquium 10/5/06

Dynamic Model for Ball-Bat Collision

AMN,Am. J. Phys, 68, 979 (2000)

- Collision excites bending vibrations in bat
- hurts! breaks bats
- dissipates energy
- lower COR, vf
- Dynamic model of collision
- Treat bat as nonuniform beam
- Treat ball as damped spring

UBC Colloquium 10/5/06

f3 = 1181 Hz

f2 = 582 Hz

f4 = 1830 Hz

frequency

time

Modal Analysis of a Baseball Bat

www.kettering.edu/~drussell/bats.html

UBC Colloquium 10/5/06

Vibrations, COR, and the “Sweet Spot”

Node of 1nd mode

+

e

vf

Evib

Strike bat here

Measure response here

UBC Colloquium 10/5/06

Independence of End Conditions

- handle moves only after ~0.6 ms delay
- collision nearly over by then
- nothing on knob end matters
- size, shape
- boundary conditions
- hands!
- confirmed experimentally

UBC Colloquium 10/5/06

q independent of end conditions:

experimental proof

Conclusion: mass added in knob has no effect on collision efficiency (q)

UBC Colloquium 10/5/06

Why Does Aluminum Outperform Wood?

- Aluminum has thin shell
- Hoop modes give “trampoline” effect
- larger COR, vf

UBC Colloquium 10/5/06

A Simple Physical Picture

- Two springs mutually compress each other
- KE PE KE
- PE shared between “ball spring” and “bat spring”
- PE in ball mostly dissipated(~80%!)
- PE in bat mostly restored
- Net effect: less overall energy dissipated
- ...and therefore higher ball-bat COR
- …more “bounce”
- Also seen in golf, tennis, …

UBC Colloquium 10/5/06

The Trampoline Effect: A Closer Look

“hoop” modes: cos(2)

Thanks to Dan Russell

“ping”

UBC Colloquium 10/5/06

The Trampoline Effect: A Closer Look

Bending Modes vs. Shell Modes

- k R4: large in barrel
- little energy stored
- f (170 Hz, etc) > 1/
- energy goes into
- vibrations

- k (t/R)3: small in barrel
- more energy stored
- f (2-3 kHz) < 1/
- energy mostly restored

to optimize….

kbat//kball small and fhoop > 1

UBC Colloquium 10/5/06

Drag: Fd = ½ CDAv2

“Magnus” or “Lift”: FL= ½ CLAv2

Fd

mg

Aerodynamics of a Baseball(in direction leading edge is turning)

CD~ 0.2-0.5

CL ~ R/v

UBC Colloquium 10/5/06

Effect of Drag and Lift on Trajectories

FL(Magnus)

Fd

mg

- drag effect is huge
- lift effect is smaller but significant

UBC Colloquium 10/5/06

Some Effects of Drag

- Reduced distance on fly ball
- Reduction of pitched ball speed by ~10%
- Asymmetric trajectory:
- Total Distance 1.7 x distance at apex
- Optimum home run angle ~30o-35o

UBC Colloquium 10/5/06

Fd

mg

Some Effects of Lift- Backspin makes ball rise
- “hop” of fastball
- undercut balls: increased distance, reduced optimum angle of home run
- Topspin makes ball drop
- “12-6” curveball
- topped balls nose-dive
- Breaking pitches due to spin
- Cutters, sliders, etc.

UBC Colloquium 10/5/06

New Experiment at Illinois

- Fire baseball horizontally from pitching machine
- Use motion capture to track ball over ~5m of flight and determine x0,y0,vx,vy,,ay
- Use ay to determine Magnus force as function ofv,

UBC Colloquium 10/5/06

Two-wheel pitching machine

Baseball with reflecting dot

Motion Capture ExperimentJoe Hopkins, Lance Chong, Hank Kaczmarski, AMNUBC Colloquium 10/5/06

Typical Motion Capture Datameasure spin, CM trajectory

CM trajectory

Note: topspin ay > g

UBC Colloquium 10/5/06

Results for Lift Coefficient CL

FL= 1/2ACLv2

S=r/v

100 mph, 2000 rpm

S=0.17

Conclusion: data qualitatively consistent (~20%)

UBC Colloquium 10/5/06

Baseball Aerodynamics:Things I would like to know better

- Better data on drag
- “drag crisis”?
- Spin-dependent drag?
- Drag for v>100 mph
- Dependence of drag/lift on seam orientation
- Is the spin constant?

UBC Colloquium 10/5/06

Oblique Collisions:Leaving the No-Spin Zone

Oblique friction spin

transverse velocity reduced

spin increased

Familiar Results:

- Balls hit to left/right break toward foul line
- Topspin gives tricky bounces in infield
- Backspin keeps fly ball in air longer
- Tricky popups to infield

demo

UBC Colloquium 10/5/06

D = center-to-center offset

Bat 100 upward

Undercutting the ball backspintrajectories

“vertical sweet spot”

UBC Colloquium 10/5/06

Putting it all Together:Can curveball be hit farther than fastball?

- Bat-Ball Collision Dynamics
- A fastball will be hit faster
- A curveball will be hit with more backspin

UBC Colloquium 10/5/06

Curveball: spin doesn’t reverse

Curveball with topspin

curveball can be hit with more backspin: WHY?

Fastball: spin must reverse

Net effect: backspin larger for curveball

UBC Colloquium 10/5/06

Can Curveball Travel Farther than Fastball?

- Bat-Ball Collision Dynamics
- A fastball will be hit faster
- A curveball will be hit with more backspin
- Aerodynamics
- A ball hit faster will travel farther
- Backspin increases distance
- Which effect wins?
- Curveball, by a hair!

UBC Colloquium 10/5/06

Work in Progress

- Collision experiments & calculations to elucidate trampoline effect
- New studies of aerodynamics
- Experiments on oblique collisions
- No data!

UBC Colloquium 10/5/06

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
- www.npl.uiuc.edu/~a-nathan/pob
- [email protected]
- Go Red Sox!

UBC Colloquium 10/5/06

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