Maths and Sport John D Barrow

Why Do Tightrope Walkers Carry Long Poles ?

High inertia Slow to move Low inertia Quick to move Mass M Average radius R mass spread C Inertia = C  Mass  (Radius) 2 Inertia high low

Large inertia means you wobble more slowly Period of wobble oscillation   (inertia)

StableUnstable

Inertia in Sports Divers Divers Skaters Skaters Tennis players Tennis players Cyclists Cyclists Runners Runners Decreasing inertia Increases rate of spin:   1/r 2

Rigging Rowing Eights

Rowing Has Its Moments +N -N F F 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 Boat s

Moment = -Ns + N(s+r) + N(s+2r) - N(s+3r) = 0 The Italian Rig No wiggle!

Four no-wiggle rigs for eights Italian tandem Rig ‘Italian tandem Rig ’ ‘German Rig’ Moment = = 0 Moment = = 0 Moment = = 0 Moment = = 0 New* New*

Canada used the German rig to win 2008 Olympics ! = 0

A balanced boat with N rowers Is only possible if N is divisible by 4 29 possible N = 12 crews 263 possible N = 16 crews 29 possible N = 12 crews 263 possible N = 16 crews 2724 possible N = 20 crews

Putting The Shot – Two Surprises World record metres

Max range isn’t achieved with 45 degree launch angle

Launching from above ground level h  2 m

R max = h tan(2  max ) 21.3 metres (= 70 ft) optimal angle is 42 deg metres (=50 ft) deg 10.7 metres (= 35 ft) deg

The Second Surprise Top class shot putters use a launch angle of about 37 deg – not deg of about 37 deg – not degBecause… They can’t achieve the same launch speed at all launch angles at all launch angles

A Constrained Optimisation Typically degrees is best But is athlete dependent Launch speed falls as angle increases

Weird Judging Means Ice Skating Ladies Figure Skating Salt Lake City Olympics

SkaterShortLongTotal Kwan Hughes Cohen Slutskaya1.0?? Before the last competitor skates… Lowest scores lead

SkaterShortLongTotal Hughes Slutskaya Kwan Cohen And after Slutskaya skates… Hughes wins by tie-break! Slutskaya has changed the order of Hughes and Kwan!

The Moral Don’t add preferences or ranks If A beats B and B beats C It doesn’t mean A will beat C Preference votes with orders ABC, BCA, CAB imply A bts B 2-1 and B bts C 2-1 but C bts A 2-1

££££££ £££££ ££££ £££ ££ £ ££££££ £££££ ££££ £££ ££ £

Naim Suleymanoglu (b.1967) lifts 3 times his body weight

Strength versus Size Weight is proportional to mass Weight is proportional to mass W = Mg  M W = Mg  M Mass = density x volume Mass = density x volume M  R 3 M  R 3 Weight is proportional to volume Weight is proportional to volume W  R 3 W  R 3

Strength versus Size Strength is proportional to area  R 2

Strength and Weight If the density of the material stays constant then If the density of the material stays constant then strength  (weight) 2/3 Strength does not keep pace with weight Strength does not keep pace with weight (strength)  (weight)  (weight) -1/3 (strength)  (weight)  (weight) -1/3 Giants eventually break! Giants eventually break!

(strength) 3  (weight) 2

The fastest and most dangerous is ….

Drag Car Racing It’s madness! 0 to 100 mph in 0.7secs ¼ of a mile in 4 seconds Top speed 330mph Accelerations of 5-6g, retinal damage Faster than a space rocket launch Ear-drum splitting sound levels

Motion at Constant Power Power, P = Force  Velocity = m  dV/dt  V = ½ d/dt(V 2 ) V 2 =2Pt if V = 0 at t = 0 from X = 0 X = (8P/9) 3/2 t 3/2 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 mV 2 = Fx =  mgx So V = V 0 = (2  mgx 0 ) 1/2 then move at constant power for x  x 0