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