Maths and Sport John D Barrow

1. Maths and Sport John D Barrow

2. Why Do Tightrope Walkers Carry Long Poles ?

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

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

7. Stable Unstable

8. Inertia in Sports • Divers • Skaters • Tennis players • Cyclists • Runners Decreasing inertia Increases rate of spin:   1/r2

10. Rigging Rowing Eights

11. Rowing Has Its Moments +N F then F -N

12. The Wiggling Boat s 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

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

14. Four no-wiggle rigs for eights ‘Italian tandem Rig ’ Moment = 1-2-3+4+5-6-7+8 = 0 New* Moment = 1-2-3+4-5+6+7-8 = 0 ‘German Rig’ Moment = 1-2+3-4-5+6-7+8 = 0 New* Moment = 1+2-3-4-5-6+7+8 = 0

15. Canada used the German rig to win 2008 Olympics ! 1-2+3-4-5+6-7+8 = 0

16. 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 2724 possible N = 20 crews

17. Putting The Shot – Two Surprises World record 23.12 metres

18. Max range isn’t achieved with45 degree launch angle

19. Launching from above ground level h  2 m

20. 21.3 metres (= 70 ft) optimal angle is 42 deg 15.24 metres (=50 ft) ------------------- 41 deg 10.7 metres (= 35 ft) -------------------- 39 deg Rmax = h tan(2max)

21. The Second Surprise Top class shot putters use a launch angle of about 37 deg – not 42-43 deg Because… They can’t achieve the same launch speed at all launch angles

22. A Constrained Optimisation Launch speed falls as angle increases Typically 34-38 degrees is best But is athlete dependent

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

24. Before the last competitor skates… Lowest scores lead

25. And after Slutskaya skates… Hughes wins by tie-break! Slutskaya has changed the order ofHughes and Kwan!

26. 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

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

31. Strength versus Size • Weight is proportional to mass • W = Mg  M • Mass = density x volume • M  R3 • Weight is proportional to volume • W  R3

32. Strength versus Size Strength is proportional to area  R2

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

34. (strength)3 (weight)2

35. The fastest and most dangerous is ….

36. 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

37. Motion at Constant Power Power, 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 =mgx So V = V0 = (2mgx0)1/2 then move at constant powerfor x  x0