The Bounce of the The Bounce of the SuperballSuperballJohn D BarrowJohn D Barrow

Putting The Shot – Two Putting The Shot – Two SurprisesSurprises

World record 23.12 metres

Max range isn’t achieved withMax range isn’t achieved with45 degree launch angle45 degree launch angle

Range depends on VRange depends on V22

Launching from above ground levelLaunching from above ground level

h 2 m

h = 2m, g = 9.8m/sh = 2m, g = 9.8m/s22, v = 14m/s, v = 14m/sreduce g or increase h or v by 1% increases range by (20, 2, 40) cmreduce g or increase h or v by 1% increases range by (20, 2, 40) cm

Rmax = h tan(2max)

21.3 metres (= 70 ft) optimal angle is 42 deg15.24 metres (=50 ft) ------------------- 41 deg10.7 metres (= 35 ft) -------------------- 39 deg

World record23.12m

The Second SurpriseThe Second Surprise

Top class shot putters use a launch angle Top class shot putters use a launch angle of about 37 deg – not 42-43 degof about 37 deg – not 42-43 deg

Because…Because…

They can’t achieve the same launch speedThey can’t achieve the same launch speed at all launch anglesat all launch angles

A Constrained OptimisationA Constrained Optimisation

Typically 34-38 degrees is bestTypically 34-38 degrees is bestBut is athlete dependentBut is athlete dependent

Launch speed falls as Launch speed falls as angle increasesangle increases

The World Goes RoundThe World Goes Round

GravityGravity Range depends on VRange depends on V22/g/g g varies with latitude because of the non-g varies with latitude because of the non-

spherical shape and rotation of the Earthspherical shape and rotation of the Earth Net g measured with a spring balance is Net g measured with a spring balance is

bigger at the Poles than at the equatorbigger at the Poles than at the equator

MgMgnetnet = Mg - Mr = Mg - Mr22

Mg(Equator) < Mg(Poles)Mg(Equator) < Mg(Poles)

200Kg in Mexico City weighs the same as 200Kg in Mexico City weighs the same as 200.8 Kg in Helsinki200.8 Kg in Helsinki

2m HJ in Helsinki is 2.05 in Mexico city. 2m HJ in Helsinki is 2.05 in Mexico city. An 8m LJ is 8.20mAn 8m LJ is 8.20m

Air Resistance is a Drag – But Air Resistance is a Drag – But ImportantImportant

Launch speed = 45 metres per sec

Range in air = 98.5 m Range in vacuum = 177.1 m

Max height in air = 53.0 m Max height in vacuum = 76.8 m

Drag on sphere = ½ Drag on sphere = ½ airair A C A Cdd v v22 Cd 0.3

Chucking Things More RealisticallyChucking Things More Realistically

Left: solid trajectory for small Left: solid trajectory for small resistance (prop to v = 10m/s ) resistance (prop to v = 10m/s ) with 45 deg launch; dotted has with 45 deg launch; dotted has launch slightly greater than 45 launch slightly greater than 45 deg and gives longer range.deg and gives longer range.

Right: large initial speed v = 300 m/s for 10 deg (solid) 20 deg (dashed) and 30 deg (dotted) angle of launch. Fall is steeper than the rise. Not a parabola now.

Projectiles with Air ResistanceProjectiles with Air Resistance

Dimples Can Give You A LiftDimples Can Give You A Lift

Dimpling decreases drag and increases lift by inducing turbulence in the boundary layer and delaying separation of the flow from the ball

Lift arises from back spin on ball. It gives greaterrelative velocity between the ball and the air at the top than the bottom. So there is lower pressure and an upward force on the ball. The flow at the top can exceed the speedneeded for turbulence in the surface (‘boundary’) layer while the flow at the bottom stays below it.A ball with top spin is pushed downwards ie ‘negative lift’.

x = K ln[1+ At]x = K ln[1+ At]

y = [y = [ K – C]ln[1 + At] + Dt – ¼ gt K – C]ln[1 + At] + Dt – ¼ gt22

Peter Tait’s (1890-3) solution Peter Tait’s (1890-3) solution for shallow launch angles (sinfor shallow launch angles (sin ))

Golf-Ball CrystallographyGolf-Ball Crystallography

Two dimple patterns with icosahedral symmetry

Catching a Moving BallCatching a Moving Ball

Move so as to maintain a constant rate of increase of the tangent of the angle of elevation

d(tan) /dt = constant!Strategy fails when air resistance is included

No airresistance

Hit straight at a fielder

ImpactsImpacts MV + mv = MU + muMV + mv = MU + mu u – U = e(V –v)u – U = e(V –v) If m is stationary If m is stationary

before impact v = 0before impact v = 0 u = MV(1+e)/(M+m)u = MV(1+e)/(M+m) U = V(M-em)/(M+m)U = V(M-em)/(M+m) Golf ball e = 0.7, m = Golf ball e = 0.7, m =

0.046 Kg, M = 0.2 Kg0.046 Kg, M = 0.2 Kg V(clubhead) = 50 m/s V(clubhead) = 50 m/s

gives u = 34 m/sgives u = 34 m/s

M m

Speed of M V U

Speed of mv u

Optimal Clubhead-to-Ball Mass Optimal Clubhead-to-Ball Mass RatioRatio

Expts: V = C/MExpts: V = C/M1/n1/n , n , n 5.3, C constant 5.3, C constant u = MV(1+e)/(M+m) = CMu = MV(1+e)/(M+m) = CM1-1/n1-1/n/(M+m)/(M+m) What is the M/m value that gives max uWhat is the M/m value that gives max u du/dM = 0: (M+m)(n-1)=nMdu/dM = 0: (M+m)(n-1)=nM M/m = n-1 for max uM/m = n-1 for max u m = 0.046 Kg and n = 5.3 m = 0.046 Kg and n = 5.3

M = 0.20 Kg M = 0.20 Kg Which is about right!Which is about right!

Energy Efficiency = ball KE/clubhead KE = 43%Energy Efficiency = ball KE/clubhead KE = 43%

The Centre of PercussionThe Centre of Percussion

Painless BattingPainless Batting The h = r + I/Mr condition for a thin The h = r + I/Mr condition for a thin

uniform rod of length 2r with uniform rod of length 2r with I = 1/3 MrI = 1/3 Mr22

h = 4r/3 = 2/3 h = 4r/3 = 2/3 (2r) (2r)

Hitting the ball 2/3Hitting the ball 2/3rdsrds of the way down of the way down the bat creates no reaction at the the bat creates no reaction at the

pivotal point on the handlepivotal point on the handle

h

Hit thru centre Slides without rolling

Hit above centreSlides and rotates

Where is the speed of sliding to the right equal to the rotational speed to the left? No slip at base contact point. Then it rebounds without sliding.

V = Ft/MV = Ft/M = slide speed = linear rotational velocity = F r t (h-r) /IF r t (h-r) /Ih = r + I/Mrh = r + I/Mr where I = 2MrI = 2Mr22/5/5 for a sphere

h = 0.7 h = 0.7 2r = 0.7 2r = 0.7 ball’s diameter ball’s diameter 3.5 cm 3.5 cmRulesRules: cushion height 0.635 + 0.10 ball’s diameter to reduce downward

wear on the table near the cushion gutter (David Alciatore)

Cushioning the BlowCushioning the Blow

Bouncing BallsBouncing Balls

No spin With spin

The SuperballThe Superball Invented by Norman Stingley in 1965 Invented by Norman Stingley in 1965

who called it the ‘Highly Resilient who called it the ‘Highly Resilient Polybutadiene Ball’ (patent 3241834)Polybutadiene Ball’ (patent 3241834)

Manufactured by Wham-OManufactured by Wham-O very high e > 0.7 Will bounce over a 3-very high e > 0.7 Will bounce over a 3-

storey building if thrown hard.storey building if thrown hard. Rough surface, reverses direction of spin Rough surface, reverses direction of spin

at each bounce at each bounce Drop 2 one above the other and the top Drop 2 one above the other and the top

one flies 9 times higherone flies 9 times higher Lamar Hunt, founder of the American Lamar Hunt, founder of the American

Football League invented the term Super Football League invented the term Super Bowl for the final match after watching Bowl for the final match after watching his children play with a Super Ballhis children play with a Super Ball

The Bounce of the SuperballThe Bounce of the Superball

Equate total energy of motion = Equate total energy of motion = ½MV½MV22 + ½I + ½I22 angular momentum about contact pt = angular momentum about contact pt = II - MRV - MRV before and after bounce before and after bounce

V(out) = -eV(in)V(out) = -eV(in) in horizontal and vertical directions in horizontal and vertical directionsNo slip at contact point –- a perfectly rough ballNo slip at contact point –- a perfectly rough ball

Normal component of velocity is reversed at collision point Normal component of velocity is reversed at collision point

Tennis ballSuperball

= 0

= +5 rev/stopspin

= 0= +2 rev/stopspin

= -10backspin

= +10 topspin

= -10backspin = +0.5

topspin

is the angular velocity (spin)is the angular velocity (spin)

20o

low-speed impact

Path of a smooth ballPath of a smooth ballInside a square boxInside a square box

Path of a rough Superball inside a square box

Superball Snooker is DifferentSuperball Snooker is Different

Happy Christmas!