Download - Maths and Sport John D Barrow
Maths and Sport
John D Barrow
Why Do Why Do Tightrope WalkersTightrope Walkers
Carry Long Poles ?Carry Long Poles ?
• High inertiaSlow to move
• Low inertiaQuick to move
Mass MAverage radius Rmass spread C
Inertia = C Inertia = C Mass Mass (Radius) (Radius)22
InertiaInertia
high low
Large inertia means you wobble more slowlyLarge inertia means you wobble more slowly
Period of wobble oscillation Period of wobble oscillation (inertia)(inertia)
StableStable UnstableUnstable
Inertia in SportsInertia in Sports
DiversDivers SkatersSkaters Tennis playersTennis players CyclistsCyclists Runners Runners
Decreasing inertiaDecreasing inertiaIncreases rateIncreases rate
of spin: of spin: 1/r 1/r22
Rigging Rowing EightsRigging Rowing Eights
Rowing Has Its MomentsRowing Has Its Moments
+N
-N
FF then
Moment on boat = -Ns + N(s+r) - N(s+2r) + N(s+3r) = +2Nr
Then, half a stroke later…N reverses to –N and..
Moment on boat = -2Nr
The Wiggling BoatThe Wiggling Boat
s
Moment = -Ns + N(s+r) + N(s+2r) - N(s+3r) = 0
The Italian RigThe Italian Rig
No wiggle!
Four no-wiggle rigs for eightsFour no-wiggle rigs for eights
‘Italian tandem RigItalian tandem Rig ’
‘‘German Rig’German Rig’
Moment = 1-2-3+4+5-6-7+8 = 0
Moment = 1-2-3+4-5+6+7-8 = 0
Moment = 1-2+3-4-5+6-7+8 = 0
Moment = 1+2-3-4-5-6+7+8 = 0
New*New*
New*New*
Canada used the German rig to win 2008 Olympics !
1-2+3-4-5+6-7+8 = 0
A balanced boat with N rowers A balanced boat with N rowers Is only possible if N is divisible by 4Is only possible if N is divisible by 4
29 possible N = 12 29 possible N = 12 crews 263 possible N crews 263 possible N
= 16 crews= 16 crews2724 possible N = 20 2724 possible N = 20
crewscrews
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
Launching from above Launching from above ground levelground level
h 2 m
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
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
Weird Judging Weird Judging Means Means
Ice SkatingIce SkatingLadies Figure Skating Ladies Figure Skating
Salt Lake City OlympicsSalt Lake City Olympics
SkaterSkater ShortShort LongLong TotalTotalKwanKwan 0.50.5 2.02.0 2.52.5
HughesHughes 2.02.0 1.01.0 3.03.0
CohenCohen 1.51.5 3.03.0 4.54.5
SlutskaySlutskayaa
1.01.0 ?? ??
Before the last competitor skates…Before the last competitor skates…
Lowest scores leadLowest scores lead
SkaterSkater ShortShort LongLong TotalTotalHughesHughes 2.02.0 1.01.0 3.03.0
SlutskaySlutskayaa
1.01.0 2.02.0 3.03.0
KwanKwan 0.50.5 3.03.0 3.53.5
CohenCohen 1.51.5 4.04.0 5.55.5
And after Slutskaya skates…And after Slutskaya skates…
Hughes wins by tie-break!Hughes wins by tie-break!Slutskaya has changed the order ofSlutskaya has changed the order of Hughes and Kwan!Hughes and Kwan!
The MoralThe MoralDon’t add preferences Don’t add preferences
or ranksor ranksIf A beats B and B If A beats B and B
beats Cbeats CIt doesn’t mean A will It doesn’t mean A will
beat Cbeat CPreference votes with orders ABC, BCA, CABPreference votes with orders ABC, BCA, CAB
imply imply A bts B 2-1 and B bts C 2-1A bts B 2-1 and B bts C 2-1
butbutC bts A 2-1C bts A 2-1
Is the Premier LeagueRandom?
20 teams each play 38 games
AssumeProb(draw) = ¼
Prob(home win) = 3/8Prob(away win) = 3/8
Now ‘play’ all the matches
The Random Premier LeagueTeam Won Lost Drawn Points Actual 1 19 10 9 67 90 Arsenal
2 18 9 11 63 79 3 18 8 12 62 75 4 17 10 11 61 60 5 16 10 12 58 56 6 16 8 14 56 56 7 13 16 9 55 53 8 15 9 14 54 53 9 16 5 17 53 52 10 15 8 15 53 50 11 15 8 15 53 48 12 14 11 13 53 47 13 13 13 12 52 45 14 14 9 15 51 45 15 13 12 13 51 44 16 15 4 19 49 41 17 11 11 16 44 39 18 9 16 13 43 33 19 9 8 21 35 33 20 8 7 23 31 33 Wolves
“Football is afunny game”
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Naim Suleymanoglu (b.1967) lifts 3 times his body weightNaim Suleymanoglu (b.1967) lifts 3 times his body weight
Strength versus SizeStrength versus Size
Weight is Weight is proportional to massproportional to mass W = Mg W = Mg M M
Mass = density x Mass = density x volumevolume M M R R33
Weight is Weight is proportional to proportional to volume volume W W R R33
Strength versus SizeStrength versus Size
Strength is proportional to area Strength is proportional to area R R22
Strength and WeightStrength and Weight
If the density of the material stays If the density of the material stays constant thenconstant then
strength strength (weight) (weight)2/32/3
Strength does not keep pace with weight Strength does not keep pace with weight (strength)(strength)(weight) (weight) (weight) (weight)-1/3-1/3 Giants eventually break!Giants eventually break!
(strength)(strength)33 (weight) (weight)22
The fastest and most dangerous is ….The fastest and most dangerous is ….
Drag Car RacingDrag Car RacingIt’s madness!It’s madness!
0 to 100 mph in 0.7secs0 to 100 mph in 0.7secs
¼ of a mile in 4 seconds¼ of a mile in 4 seconds
Top speed 330mphTop speed 330mph
Accelerations of 5-6g, retinal damageAccelerations of 5-6g, retinal damage
Faster than a space rocket launchFaster than a space rocket launch
Ear-drum splitting sound levelsEar-drum splitting sound levels
Motion at Constant PowerMotion at Constant PowerPower, P = Force Velocity = m dV/dt V = ½ d/dt(V2)
V2 =2Pt if V = 0 at t = 0 from X = 0
X = (8P/9)3/2 t3/2
V = (3PX/m)1/3
‘Huntingdon’s Rule’Speed(Mph) = K (Power in horsepower/mass in
pounds)1/3
K(calculated) = 270 >> K(observed) = 225
But traction is limited at start ½ mV2 = Fx =mgxSo V = V0 = (2mgx0)1/2 then move at constant power for x x0