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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!