Citius, Altius, Fortius Records, Medals and Drug Testing John D. Barrow

AthleteInd’lG-S-BTotalG+S+BTotal3G+2S+BInd’l+teamG-S-BGrandTotalG+S+B Grand Total 3G+2S+B Phelps * Ewry Caslavska Lewis Latynina * Andianov Nurmi Shakhlin …. 57 th Redgrave The Most Successful Summer Olympians

G+S+B =M  P 0.73  GDP 0.27 Big variances though Pop in millions GDP in billions USD

1992 Barcelona Games Best Fit S = 3G + 2S + B S = 0.22  P 1/3  G 2/3 P = population in millions G = GDP in billions USD Roughly S = (PG 2 ) 1/3 or S’ = 0.5S 3 = ½ PG 2 = ‘energy of a nation’

Peak to peak periodicity of 4T How to Stop Money Killing Competition Ordinary diff eqn Delay eqn Exact soln S(t) = e Ft Model for the ‘Draft’ system in US professional sport Bottom team last year gets first pick of new players Top team gets last pick Premier league model

Score = 1 for win, ½ for draw, 0 for loss With T = 2 yr delay there is a clear 8 yr cycle : 8.24 yr for whole NFL 8.24 yr for whole NFL Bufffalo Bills 8.3 yr period What Goes Up Must Come Down

Strategy for Success Invest in sports that few countries do Invest in sports that few countries do Pick events with lots of relays, and team awards and ‘double bronzes’ Pick events with lots of relays, and team awards and ‘double bronzes’ Pick sports with strong inter-event similarity (like cycling) Pick sports with strong inter-event similarity (like cycling) Pick team sports where training and organisation can dominate Pick team sports where training and organisation can dominate Don’t do too many different sports Don’t do too many different sports Encourage some to change events Encourage some to change events

The Triathlon is Biased Winner spent 16.7% of his time swimming, 28.3% running, 0.8% in transitions, and 54.2% cycling 0.8% in transitions, and 54.2% cycling !

The Equitempered Triathlon Roughly equal time on each discipline Current event 1.5 km swim + 40 km bike ride +10 km run 1.5 km swim + 40 km bike ride +10 km run Better to keep the total time roughly the same – say 1 hr 48m with equal 36 min time on each of the three stages New event should be 3 km swim + 24 km bike ride + 12 km run A Major Improvement !

Decathlon Running: 100m + 110m hurdles + 400m m Throwing: shot + discus + javelin Jumping: high jump + long jump + pole vault Times and distances  points scores Heptathlon: 200m + 800m + HJ + 100H + 800m + Javelin + shot put

Power-Law Scoring Run Jump Throw Distance achieved = D Time run = T C>1 ‘progressive’ C = 1 ‘neutral’ C<1 ‘regressive’ IAAF 2001 Tables Fix A, B, C

Heptathlon 200m C = m C = H C = HJ C = LJ C = 1.41 SP C = 1.05 JT C = 1.04

All-time top 100 performance patterns What you need to do to Earn 900 points in each event for a 9000 points total Which Events Give the Biggest Pay-off ? World record 9026 points Roman Sebrle in 2001 LJ, 110H, sprints ** 1500m, throws xx

Could avoid points transformations by having B-Total = LJ  HJ  PV  JT  DT  SP_______ T(100)  T(400m)  T(100mH) T(1500m) List all distances in metres and all times in seconds Units of total are m 6 /s 4 A Different Scoring Scheme It has its own biases! Sebrle (9026 pts): B = 2.29 Dvorak (8994 pts): B = 2.40

Best Evers and What Ifs If scoring is made highly progressive C = 2 for all events Dvorak (world no 2) becomes new world record holder with 9468 This change dramatically favours the throwers C = 1.1  2 Daley Thompson missed world record by 1 pt in 1984 Olympics Subsequent change in scoring tables gave him the record! Current world decathlon record = 9026 World record in every event = 12,500 Best ever decathlon performances in each event = 10,485 Usain Bolt’s 9.58s 100m record = 1202 Fastest 100m in a decathlon = 10.22s = 1042 ‘Best’ world record is 74.08m discus (Schult) = 1383

Random Records Random Records In year 1 the result must be a record. So the number of record years is In year 1 the result must be a record. So the number of record years is1 In year 2, if the result is independent of year 1, there is a chance of 1/2 of beating the record from year 1 and a chance of 1/2 of not beating it. So the expected number of record years in the first 2 years is In year 2, if the result is independent of year 1, there is a chance of 1/2 of beating the record from year 1 and a chance of 1/2 of not beating it. So the expected number of record years in the first 2 years is 1 + 1/2 In year 3 there are just two ways in which the 6 possible rankings In year 3 there are just two ways in which the 6 possible rankings (123, 132, 321, 213, 312, 231) of the results in years 1, 2 and 3 could produce a record in year 3 (ie a 1 in 3 chance). So the expected number of record years after 3 years is of the results in years 1, 2 and 3 could produce a record in year 3 (ie a 1 in 3 chance). So the expected number of record years after 3 years is 1 + 1/2 + 1/3 If you keep on going, applying the same reasoning to each new year, you will find that after n independent years the expected number of record years is the sum : If you keep on going, applying the same reasoning to each new year, you will find that after n independent years the expected number of record years is the sum : 1 + 1/2 + 1/3 + 1/ /n = H(n) The ‘harmonic’ series

But H goes to infinity very slowly H(n) = 1 + 1/2 + 1/3 + 1/4 + 1/5 + …. + 1/n Then H(1) = 1, H(2) = 1.5, H(3) = 1.833, H(4) = 2.083, H(10) = 2.93, H(100) = 5.19 The sum grows very slowly as the number of terms increases: H(256) = but H(1000) = 7.49 H(1,000,000) = When n gets large H(n) only increases as fast as the natural logarithm of n and approximately H(n) = log e n

World men’s 100m record frequency We expect H(100)  5 records since 1910 and H(26)  4 Olympic records So record-setting is not a random process

New events Women’s Athletics Records are Very Rare

11 women’s world records are more than 20 yrs old 2 men’s records are more than 20 years old are more than 20 years old

Drug Testing Stats Take Drugs (1%) No Drugs (99%) Test + 80%9.6% Test - 20%90.4% Assume 1% of athletes take drugs and 80% of tests detect their presence But 9.6% of tests are positive when no drugs are present (‘false positives’)

What If You Test Positive? Took Drugs (1%) No Drugs (99%) Test + 1%  80% = true + = true + 99%  9.6% = false + Test - 1%  20% = false - 99%  90.4% = true - What is the chance you took drugs? Prob you are taking drugs if test + is 0.008/( ) = Only detect 7.8% of drug users because we have 9.5% false positives

Thomas Bayes’ Theorem P(A|B) = P(B|A)P(A) = _________ P(B|A)P(A) ______ P(B) P(B|A)P(A) + P(B|notA)P(notA) Prob of Taking drugs if test + 1% 80% 1% False + 9.6% 99% = 7.8% A = taking drugs B = test + not A = not taking drugs

Race Walking Start of the 3500 metre walk at the 1908 Summer Olympics

‘Lifting’ in Race Walking ‘Race walking is a progression of steps so taken that the walker makes contact with the ground so that no visible (to the human eye) loss of contact occurs.’ USTFA rules The human eye records at only about 25 frames per sec

This film is of an experienced Italian male who, according to the IAAF, has walked a 20K race in 1:23:55 ‘Three warnings from different judges and ‘Three warnings from different judges and you are out’ disqualification rule’

Jefferson Perez (Ecuador) 1996 Olympic 20K champion during a 10K race. Note the lack of up down movement of his centre of gravity He averaged just over 6 minutes per mile. He is taking about 186 steps/min. This film is actual speed

The Forces That Are With You Stride length S Leg length L Weight Mg V = 2Ld/dt   Vertical accn R-Mg = M d 2 (2Lcos)/dt 2 Rotational accn 4ML 2 d 2 /dt 2 = 2MgLsin

g = 9.8 m/s 2, L = 1m, S = 1.3m then max V before contact is lost is V 2 = ½ gL[3(4 - S 2 /L 2 ) – 4] V max = 1.7 m/s World record speed over 20 Km is V > 4m/s Reduce S to 1m still gives V max = 2.42 m/s Increase the effective leg length using hip sway ? Lifting Seems Inevitable

Hip Sway can give L eff  S Need 0.7m hip extension !!