Cover
About the Book
About the Author
Also by John D. Barrow
Dedication
Title Page
Epigraph
Preface
1. How Usain Bolt Could Break His World Record With No Extra Effort
2. All-rounders
3. The Archers
4. The Flaw of Averages
5. Going Round the Bend
6. A Question of Balance
7. Anyone for Baseball, Tennis or Cricket?
8. Bayes Watch
9. Best of Three
10. High Jumping
11. Having the Right Birthday
12. Air Time
13. Kayaking
14. Do You Need a Cox?
15. On the Cards
16. Wheels on Fire
17. Points Scoring
18. Diving
19. The Most Extreme Sport of All
20. Slip Slidin’ Away
21. Gender Studies
22. Physics for Ground Staff
23. What Goes Up Must Come Down
24. Left-handers versus Right-handers
25. Ultimate Pole-vaulting
26. The Return of the Karate Kid
27. Leverage
28. Reach for the Sky
29. The Marathon
30. All That Glitters Is Not Gold
31. Don’t Blink First
32. Ping-pong is Coming Home
33. A Walk on the Wild Side
34. Racing Certainties
35. What is the Chance of Being Disqualified?
36. Rowing Has Its Moments
37. Rugby and Relativity
38. Run Rates
39. Squash – A Very Peculiar Practice
40. Faking It
41. A Sense of Proportion
42. Cushioning the Blow
43. Breaststrokers
44. That Crucial Point
45. Throwin’ in the Wind
46. The Two-headed League
47. What a Racket
48. Size Matters
49. A Truly Weird Football Match
50. Twisting and Turning
51. The Wayward Wind
52. Windsurfing
53. Winning Medals
54. Why Are There Never World Records in Women’s Athletics?
55. The Zigzag Run
56. Cinderella Sports
57. Wheelchair Racing
58. The Equitempered Triathlon
59. The Madness of Crowds
60. Hydrophobic Polyurethane Swimsuits
61. Modern Pentathlon
62. Keeping Cool
63. Wheelchair Speeds
64. The War on Error
65. Matters of Gravity
66. Googling in the Caribbean
67. The Ice-skating Paradox
68. Throwing the Discus
69. Goal Differences
70. Is the Premier League Random?
71. Fancy Kit – Does It Help?
72. Triangles in the Water
73. The Illusion of Floating
74. The Anti-Matthew Effect
75. Seeding Tournaments
76. Fixing Tournaments
77. Wind-assisted Marathons
78. Going Uphill
79. Psychological Momentum
80. Goals, Goals, Goals
81. Total Immersion
82. The Great British Football Team
83. Strange But True
84. Blade Runner
85. Pairing People Up
86. Ticket Touts
87. Skydiving
88. Running High
89. The Archer’s Paradox
90. Bend It Like Beckham
91. Stop-go Tactics
92. Diving is a Gas
93. Spring is in the Air
94. The Toss of the Coin
95. What Sports Should Be in the Olympics?
96. The Cat Paradox
97. Things That Fly Through the Air With the Greatest of Ease
98. Some Like It Hot
99. The Bounce of the Superball
100. Thinking Inside the Box
Notes
Copyright
What can maths tell us about sports?
100 Essential Things You Didn’t Know You Didn’t Know About Sport sheds new light on the mysteries of running, jumping, swimming and points-scoring across the whole sporting spectrum. Whether you are a competitor striving to go faster or higher, or an armchair enthusiast wanting to understand more about your favourite sport, this is a fascinating read. Find out why high-jumpers use the Fosbury Flop; how Usain Bolt could break his records without running any faster; what is the best strategy for taking football penalties; what are the effects of those banned skin-tight swimsuits; and last but not least, why the bounce of a Superball seems to defy Newton’s laws of motion.
Written for anyone interested in sport or simple mathematics, this book will enrich your understanding of sport and enliven your appreciation of maths.
John D. Barrow is Professor of Mathematical Sciences and Director of the Millennium Mathematics Project at Cambridge University, Fellow of Clare Hall, Cambridge, a Fellow of the Royal Society, and the current Gresham Professor of Geometry at Gresham College, London. He is the bestselling author of 100 Essential Things You Didn’t Know You Didn’t Know.
ALSO BY JOHN D. BARROW
Theories of Everything
The Left Hand of Creation
(with Joseph Silk)
L’Homme et le Cosmos
(with Frank J. Tipler)
The Anthropic Cosmological Principle
(with Frank J. Tipler)
The World within the World
The Artful Universe
Pi in the Sky
Perché il mondo è matematico?
Impossibility
The Origin of the Universe
Between Inner Space and Outer Space
The Universe that Discovered Itself
The Book of Nothing
The Constants of Nature: From Alpha to Omega
The Infinite Book: A Short Guide to the Boundless, Timeless and Endless
New Theories of Everything
Cosmic Imagery: Key Images in the History of Science
100 Essential Things You Didn’t Know You Didn’t Know
The Book of Universes
TO MAHLER
who can already run
and soon will count
‘Heck, gold medals, what can you do with them’
Eric Heiden
In this Olympic year I have taken the opportunity to demonstrate some of the unexpected ways in which simple mathematics and science can shed light on what is going on in a wide range of sporting activities. The following chapters will look into the science behind aspects of human movement, systems of scoring, record breaking, paralympic competition, strength events, drug testing, diving, riding, running, jumping and throwing. If you are a coach or a competitor you may get a glimpse of how a mathematical perspective can enrich your understanding of your event. If you are a spectator or commentator then I hope that you will develop a deeper understanding of what is going on in the pool, gymnasium or stadium, on the track or on the road. If you are an educator you will find examples to enliven the teaching of many aspects of science and mathematics, and to broaden the horizons of those who thought that mathematics and sport were no more than a timetable clash. And if you are a mathematician you will be pleased to discover how essential your expertise is to yet another area of human activity. The collection of examples you are about to read covers a great many sports and tries to pick topics that have not been discussed extensively before. Occasionally, there is a little bit of Olympic history for perspective, but it is balanced by chapters about several non-Olympic sports as well, and if you wish to delve deeper with your reading or push a calculation further there are notes to show you where to begin.
I would like to thank Katherine Ailes, David Alciatore, Philip Aston, Bill Atkinson, Henry Baker, Melissa Bray, James Cranch, Marianne Freiberger, Franz Fuss, John Haigh, Jörg Hensgen, Steve Hewson, Sean Lip, Justin Mullins, Kay Peddle, Stephen Ryan, Jeffrey Shallit, Owen Smith, David Spiegelhalter, Ian Stewart, Will Sulkin, Rachel Thomas, Roger Walker, Peter Weyand and Peng Zhao for the help, discussions and useful communications that helped this book come into being. A few of the topics covered here have been presented in lectures at Gresham College in London and as part of the Millennium Mathematics Project’s activity for the London 2012 Olympics. I am most grateful to these audiences for their interest, questions and input. I must also thank family members, Elizabeth, David, Roger and Louise for their enthusiasm – although it turned to disbelief when they realised that this book wasn’t going to help them get any Olympic tickets.
John D. Barrow, Cambridge 2012.
USAIN BOLT IS THE best human sprinter there has ever been. Yet, few would have guessed that he would run so fast over 100m after he started out running 400m and 200m races when in his mid teens. His coach decided to shift him down to running 100m one season so as to improve his basic sprinting speed. No one expected him to shine there. Surely he is too big to be a 100m sprinter? How wrong they were. Instead of shaving the occasional hundredth of a second off the world record, he took big chunks out of it, first reducing Asafa Powell’s time of 9.74s down to 9.72 in New York in May 2008, and then down to 9.69 (actually 9.683) at the Beijing Olympics later that year, before dramatically reducing it again to 9.58 (actually 9.578) at the 2009 Berlin World Championships. His progression in the 200m was even more astounding: reducing Michael Johnson’s 1996 record of 19.32s to 19.30 (actually 19.296) in Beijing and then to 19.19 in Berlin. These jumps are so big that people have started to calculate what Bolt’s maximum possible speed might be. Unfortunately, all the commentators have missed the two key factors that would permit Bolt to run significantly faster without any extra effort or improvement in physical conditioning. ‘How could that be?’ I hear you ask.
The recorded time of a 100m sprinter is the sum of two parts: the reaction time to the starter’s gun and the subsequent running time over the 100m distance. An athlete is judged to have false-started if he reacts by applying foot pressure to the starting blocks within 0.10s of the start gun firing. Remarkably, Bolt has one of the longest reaction times of leading sprinters – he was the second slowest of all the finalists to react in Beijing and third slowest in Berlin when he ran 9.58. Allowing for all this, Bolt’s average running speed in Beijing was 10.50m/s and in Berlin (where he reacted faster) it was 10.60m/s. Bolt is already running faster than the ultimate maximum speed of 10.55m/s that a team of Stanford human biologists recently predicted for him.1
In the Beijing Olympic final, where Bolt’s reaction time was 0.165s for his 9.69 run, the other seven finalists reacted in 0.133, 0.133, 0.134, 0.142, 0.145, 0.147, 0.165 and 0.169s.
From these stats it is clear what Bolt’s weakest point is: he has a very slow reaction to the gun. This is not quite the same as having a slow start. A very tall athlete, with longer limbs and larger inertia, has got more moving to do in order to rise upright from the starting blocks.2 If Bolt could get his reaction time down to 0.13, which is very good but not exceptional, then he would reduce his 9.58 record run to 9.56. If he could get it down to an outstanding 0.12 he is looking at 9.55 and if he responded as quickly as the rules allow, with 0.1, then 9.53 is the result. And he hasn’t had to run any faster!
This is the first key factor that has been missed in assessing Bolt’s future potential. What are the others? Sprinters are allowed to receive the assistance of a following wind that must not exceed 2m/s in speed. Many world records have taken advantage of that and the most suspicious set of world records in sprints and jumps were those set at the Mexico Olympics in 1968 where the wind gauge often seemed to record 2m/s when a world record was broken. But this is certainly not the case in Bolt’s record runs. In Berlin his 9.58s time benefited from only a modest 0.9m/s tailwind and in Beijing there was nil wind, so he has a lot more still to gain from advantageous wind conditions. Many years ago, I worked out how the best 100m times are changed by wind.3 A 2m/s tailwind is worth about 0.11s compared to a nil-wind performance, and a 0.9m/s tailwind 0.06s, at a low-altitude site. So, with the best possible legal wind assistance and reaction time, Bolt’s Berlin time is down from 9.53s to 9.47s and his Beijing time becomes 9.51s. And finally, if he were to run at a high-altitude site like Mexico City, then he could go faster still and effortlessly shave off another 0.07s.4 So he could improve his 100m time to an amazing 9.4s without needing to run any faster.5
HUMANS ARE OFTEN compared rather unfavourably with the champions of the animal kingdom: cheetahs sprinting faster than the motorway speed limit, ants carrying many times their body weight, squirrels and monkeys performing fantastic feats of aerial gymnastics, seals that swim at superhuman speeds, and birds of prey that can pluck pigeons out of the air without the need for guns. It is easy to feel inadequate. But really we shouldn’t. All these stars of the animal kingdom are really nowhere near as impressive athletes as humans. They are very good at very special things and evolution has honed their ability to dominate their competitors in a very particular niche. We are quite different. We can swim for miles, run a marathon, run 100m in less than ten seconds, turn a somersault, ride a bike or a horse, high jump over eight feet, shoot accurately with rifles and bows, throw small objects nearly a hundred metres, ride a bicycle for hundreds of kilometres, row a boat, and lift much more than our body weight over our heads. Our range of physical prowess is exceptional. It’s easy to forget that no other living creature can match us for the diversity of our physical abilities. We are the greatest multi-eventers on earth.
OLYMPIC ARCHERY IS a dramatic participation sport but it is not so easy to see what is happening without a good pair of binoculars or big video monitors to replay the shots. The archers shoot seventy-two arrows at a circular target 70m away. The target is 122cm in diameter and divided into ten concentric rings, each of which is 6.1cm wide.
The two inner rings are gold and arrows landing there score 10 and 9 points. Going outwards the next two are red and score 8 and 7 points; the next two are blue and score 6 and 5; the next two are black and score 4 and 3; the last two are white and score 2 and 1. If you hit the target further out than this (or miss it completely) you score zero. These coloured circles are printed on a 125cm × 125cm square of paper that is backed by a protective layer to stop the arrows from penetrating through it.
The world’s best archer is the South Korean woman Park Sung-Hyun. She scored a total of 682 points from seventy-two arrows to win individual and team gold medals at the 2004 Athens Olympics.1 If she only scored 10s and 9s with all her arrows we can work out how she would have achieved that score. If T arrows scored 10 and the other 72–T arrows scored 9 then we know that 10T + 9(72–T) = 682 and so T = 34 gives the number of 10s scored. The number of 9s would have been 72–34 = 38. If she only scored 10s, 9s and 8s you might like to show that she must have scored thirty-five 10s, thirty-six 9s and one 8.
The difficulty of getting a particular score with one arrow is determined by the area of the annular ring that you have to hit to obtain it. The outer radii (in centimetres) of each of the ten circular rings are 6.1, 12.2, 18.3, 24.4, 30.5, 36.6, 42.7, 48.8, 54.9 and 61. Since the area of a circle is just π (= 3.14) times the square of its radius we can work out the area of each annular ring by subtracting the area of its inner bounding circle from the area of its outer bounding circle. So, for example, the area of the ring in which arrows score 9 points is π (12.22 – 6.12) = π × 6.1 × 18.3 = 350.7. I won’t work out the areas of all the target rings but the same principle gives them very easily. Now, the likelihood of your arrow gaining a particular score is given by the fraction of the target area occupied by that part of the target. The area of the whole circular target is π × 612 = 11689.9sq cm and so the probability of scoring a 9 with a randomly shot arrow that hits the target area is given by the ratio of the area in which you score 9 to the total area and this is 350.7/11689.9 = 0.03, or 3%. If I do these sums for the relative areas of all the scoring rings I get the probabilities that randomly shot arrows will hit any one of them. There is a simple pattern. The probabilities rise by 2% per ring as you move outwards through the rings. The hardest to hit is the centre ring with a 1% (i.e. 0.01) chance for a random shot; the easiest is the outer ring with a 19% (i.e. 0.19) chance of scoring 1 point.
If we add up all these average contributions we get 3.85 as the score we are likely to get from shooting a single arrow randomly at the target. If we shoot seventy-two arrows randomly then the average score we will get will be seventy-two times this, or 277, to the nearest round number. As you might expect this is far, far less than the world record score of 682. A score of 277 is what you would achieve with a purely random shooting strategy with no skill at all (except to hit some part of the target).
In calculating this we assumed that a random archer always hits the circular target. Suppose that they are not even that accurate and end up hitting anywhere at random inside the 125cm × 125cm square on which the target is printed. Its area is 15,625sq cm and you score zero if you hit this square beyond the outer circle of radius 61cm. In this case, all the overall probabilities and scores are reduced by a factor equal to the ratio of the area of the outer circle divided by the square, which is 11689.9 ÷ 15625 = 0.75. Therefore the average score obtained by shooting seventy-two arrows at random within the bounding square falls to 207.4.
If you want to test your arithmetic then you can apply exactly the same principles to calculate what score would be obtained by a random darts player. You should find that the average score is 13 points per dart, giving a score of 39 for three darts.2
AVERAGES ARE FUNNY things. Ask the statistician who drowned in a lake of average depth equal to 3cm. Yet, they are so familiar and seemingly so straightforward that we trust them completely. But should we? Let’s imagine two cricketers. We’ll call them, purely hypothetically, Anderson and Warne. They are playing in a crucial Test match which will decide the outcome of the series. The sponsors have put up big cash prizes for the best bowling and batting performances in the match. Anderson and Warne don’t care about batting performances – except in the sense that they want to make sure there aren’t any good ones at all on the opposing side – and are going all out to win the big bowling prize.
In the first innings Anderson gets some early wickets but is taken off after a long spell of very economical bowling and ends up with figures of 3 wickets for 17 runs, an average of 5.67. Anderson’s side then have to bat and Warne is on top form, taking a succession of wickets for final figures of 7 for 40, an average of 5.71 runs per wicket taken. Anderson therefore has the better (i.e. lower) bowling average in the first innings, 5.67 to 5.71.
In the second innings Anderson is expensive at first, but then proves to be unplayable for the lower-order batsmen, taking 7 wickets for 110 runs, an average of 15.71. Warne then bowls at Anderson’s team during the last innings of the match. He is not as successful as in the first innings but still takes 3 wickets for 48 runs, for an average of 16. So, Anderson has the better average bowling performance in the second innings as well, this time by 15.71 to 16.
Who should win the bowling man-of-the-match prize for the best figures? Anderson had the better average in the first innings and the better average in the second innings. Surely, there is only one winner. But the sponsor takes a different view and looks at the overall match figures. Over the two innings Anderson took 10 wickets for 127 runs for an average of 12.7 runs per wicket. Warne, on the other hand, took 10 wickets for 88 runs and an average of 8.8. Warne clearly has the better average and wins the bowling award despite Anderson having a superior average in the first innings and in the second innings!
HAVE YOU EVER wondered whether it’s best to have an inside or an outside lane in track races like the 200m where you have to sprint around the bend? Athletes have strong preferences. Tall runners find it harder to negotiate the tighter curve of the inside lane than that of the gentle outer lanes. The situation is even more extreme when sprinters race indoors where the track is only 200m around, so the bends are far tighter and the lanes are reduced in width from 1.22m to 1m. This was such a severe restriction that it became common for the athlete who drew the inside lane for the final (by being the slowest qualifier on times) to scratch from the final in indoor championships because there was so little chance of winning from the inside and a considerable risk of injury. As a result, this event has largely disappeared from the indoor championship roster.
But what about the outdoor situation where the curve is not so extreme? Most athletes don’t like to be right on the outside because you can’t see anyone (unless they pass you) for the first half of the race and you can’t run ‘off’ their pace. On the inside you have a metal kerb marking the inside of your lane and you tend not to get as close to it as you would to the simple white painted line that marks the inside of the other lanes. Generally, the fastest qualifiers from the previous round are placed in the centre two or three lanes – a clear signal that they might be advantageous. A runner’s physique is a factor too. If you are tall and long-legged you will have a harder time in the inner lanes and may have to chop your stride or run towards the outside of your lane to run freely. Potentially even more significant is the wind. If the wind is blowing at right angles to the finishing straight, into the faces of the runners when they run around the bend, then you will want to be in the outside lane so that you will be starting some way around the bend and will not have to run directly into the wind for so long – unlike those runners on the inside.
Finally, it is easy to show that you need to work harder if you run in the inside lanes. The two bends of an athletics track are semicircular. The radius of the circle traced by the inner line of the inside lane is 36.5m and each lane is 1.22m wide. So, the radius of the circle that you run in gets larger and the extra force that you have to exert to run in a circular path gets smaller and you actually run a smaller part of a circle as well. The radius of the circle traced by lane eight is 36.5 + (7 × 1.22) = 45.04m. The force needed for a runner of mass m to run in a circular path of radius r at speed v is mv2/r, so as r gets larger,1 and the bend is less tight, the force needed to maintain a given speed v decreases. If two identical runners, one in lane one and the other in lane eight, exert the same force over the first 100m of a 200m race, then the runner in lane one will have achieved a speed that is about 0.9 of that achieved in lane eight and the runner in lane eight will take 0.9 of the time. This is a very large factor – worth a whole second off the time for the first half of the race if you are running a 20s time for 200m. In practice there isn’t such a large systematic advantage to running in the outside lanes and the runner only has to supply a fraction of the full circular motion force to sprint around the curve.2
If this simple model were complete then all 200m runners would run their best times from the outside lane. In practice most records are set from lanes three and four. Even this fact is slightly biased because the fastest qualifiers for the finals of big championships will have been put in those lanes. Presumably, the psychological and tactical advantages of being able to see your opponents and judge your speed against them from an inside lane helps outweigh the mechanical advantage of running around a gentler curve.
A good final comparison to make which illustrates the effect of the curve on 200m runs is to compare the world records run on a straight track with those around a curve. Straight 200m tracks are very rare now. There used to be one at the old Oxford University track at Iffley Road (where Roger Bannister ran the first sub-four-minute mile in 1954) that was still there when I began as a student in 1974 but had been removed by the time I graduated in 1977. When Tommie Smith set his world 200m record of 19.83s around a curve at altitude in the 1968 Mexico Olympics, he had already run a remarkable3 19.5s on a straight cinder track in San Jose in 1966. This latter record was only beaten by Tyson Gay, who ran 19.41s at the Birmingham City Games in 2010, watched by a 65-year-old Smith. Gay’s fastest time around a curve is 19.58s. These time differences show the considerable slowing that is created by negotiating the curve. You might be lucky and have the wind behind you all the way in a straight 200m, but nonetheless runners find it strange to sprint such a long way without the reference points of the curve and other runners to dictate where they are and how they should apportion their effort.
IF THERE IS one attribute that is invaluable in just about any sport, it is balance. Whether you are a gymnast on the beam, a high-board diver, a spinning hammer thrower, a rugby forward snaking through the opposition’s defence, a wrestler, a judoka trying to throw an opponent or a fencer lunging forwards, it is all about balance. Try a little experiment to see how well balanced you are and get a feeling for the muscle control behind it. Just stand completely still with one foot immediately in front of the other, so that the heel of your left foot touches the toe of your right foot. You can shift your weight so that it is mainly over the front foot or the back foot but keep your hands by your sides. You will probably find that standing completely still in a relaxed way is surprisingly difficult and your calf muscles are being tensed this way and that all the time. If you spread your arms out sideways you will find it much easier to balance. But now try leaning to one side. You won’t lean very far before you lose your balance completely. Now, if you move your feet apart, in a normal standing position so they are not one behind the other in a straight line, then you will find it easier still, even with your arms by your sides – this, after all, is probably your usual stance. Lastly, go back to that difficult position with one foot directly in front of the other, but slowly crouch down low. You will find that balancing gets easier as you get nearer to the ground.
These little exercises reveal some simple principles for maintaining a good balance:
Make sure that the vertical line through your body’s centre of gravity doesn’t fall outside the base of support created by your feet. Once it does, you will fall away from equilibrium. You can experiment for yourself to see how far you can lean sideways, while keeping the body straight, before you start falling. The high-board diver will often use this instability in order to initiate his dive, leaning forward until his movement is taken over by gravity.
Broaden your base of support as much as possible. This makes it harder for your centre of gravity to fall outside your base. If you can stand on two feet, rather than on one, this will always help.
Keep your centre of gravity as low as possible. This is why you often see female gymnasts on the beam going into a low crouch position during a swing, perhaps with only one foot on the beam and one leg dangling below the beam – this lowers the centre of gravity even more. Sit astride the beam and you will see that balance is easy – your centre of gravity can’t get much lower.
Spread your weight as far from your centre as possible. This is what was happening when you spread your arms out sideways. This is changing the distribution of your mass. By moving more of it far from your centre you are increasing your inertia, or your tendency not to move. Increasing your inertia in this way won’t stop you wobbling but, crucially, it will make you wobble more slowly.1 This gives you more time to take corrective action, shift your centre of gravity sideways or downwards, as required. This is why tightrope walkers carry long poles: they are ensuring that they wobble more slowly and have more time to correct a dangerous imbalance. Without that helpful pole, the man walking between skyscrapers on a high wire would surely fall to his death once he started to wobble in the breeze.
Watch wrestling and judo, where competitors are constantly trying to make their opponents lose their balance in subtle ways, or by using their strength to force them to violate one of the principles we have highlighted.
A LOT OF people spend a lot of time hitting or chasing small spherical projectiles while dressed in unusual items of clothing. Games like baseball, tennis and cricket involve someone receiving one of these projectiles at very high speed. They have a split second to respond, either by getting out of the way, or hitting the projectile back as skilfully as they can. Which of these three sports requires the quickest reactions?
In each case the ball is different in size and can be launched by the pitcher, server or bowler at different speeds. Baseball has the simplifying feature that the ball only flies through the air, whereas in cricket and tennis it will hit the ground and rebound unpredictably because of its spin. In all three cases, the ball can swerve in the air to deceive the receiver in many ways. Let’s ignore these extra degrees of difficulty and just focus on how quickly the receiver has to react to the incoming ball in each of these three games.
First, take cricket: a cricket pitch is 22 yards (= 20.12m) long.1 The fastest bowlers achieve speeds exceeding 100mph, which is about 45m/s. The bowler will generally take a lengthy approach run in order to build up speed but the ball must be released with a straight arm or a ‘no-ball’ will be called for ‘throwing’. If the batsman stands 1m in front of the wicket then he has 19.12/45 = 0.42s before the ball arrives at his bat.
By contrast, the baseball pitcher takes no run-up. He winds himself up on the spot from one of two allowed positions, the ‘windup’ or the ‘stretch’. The counterpart of the fast bowler in cricket is the power pitcher. He relies on speed of delivery to outwit the hitter, who is 18.44m away from the pitcher when the ball is launched towards him. The quickest fastball that the best pitchers can deliver moves at about 100mph (= 45m/s). Unlike in cricket, bent arm pitching is allowed. The hitter’s reaction time is therefore just 18.44/45 = 0.41s – a bit less than the cricket batsman gets.
What about the tennis player? Over time, racket technology has led to faster and faster serves to such an extent that top-class tennis is in danger of becoming dominated by service aces with few rallies. The record books list the fastest recorded serve by a man as 163.6mph (73m/s) by Bill Tilden in 1931. How this was measured I don’t know. More reliable might be the value of 149mph (67m/s) attributed to Greg Rusedski at Indian Wells in 1998. The fastest recorded by a woman is 128mph (58m/s) by Venus Williams in 1998. The tennis court has a length of 78ft and for singles a width of 27ft. If the server and the receiver are located in opposite corners at the edges of the court then the distance travelled by the ball (neglecting differences in height above ground) is given by the length of the diagonal of a right-angled triangle with sides 78 and 27. Using Pythagoras’ theorem, this is the square root of 6,084 + 729, which is 82.54ft, or 25.16m. If we assume that a top-flight serve is hit at 140mph (62.6m/s) and ignore any loss of speed when it bounces in the service court area then the receiver has about 25.16/62.6 = 0.40s to react.
The most interesting thing about our three rough calculations is not whether baseball players react one or two hundredths of a second quicker than tennis players or cricketers. Rather, it is the striking similarity between the required reaction times in the three different sports, to within a few hundredths of a second. Each has pushed the human response quite close to its limit.
CHANCE AND PROBABILITY play a major part in our lives. From court decisions about multiple cot deaths and DNA matches, to health and safety risks, you can’t escape them. Determining likelihood is controversial and for the unwary beset by subtle pitfalls. Major miscarriages of justice have occurred because of ignorance by ‘expert’ witnesses in life and death judicial proceedings. The stakes are almost as high in the world of sport. Failing a drugs test can end your career and rob you of records, championships and multimillion-dollar commercial contracts. Thus testing athletes in a foolproof way is very important. There have been cases of incompetence by certified testing laboratories that have wrecked the careers of athletes and, in the case of Diane Modahl from 1994–8, brought about the collapse of the British Athletics Federation.
Baseball is an interesting case. Leading players are suspected of achieving their record-breaking feats by means of systematic steroid use. There is no random drug testing and disqualification system in place in US baseball, but anonymous testing has consistently revealed a worrying level of steroid use. Let’s suppose that such a test reveals that 5% of players are steroid users and scientists tell us that the test is 95% accurate. What does that mean?
Suppose that 1,200 players are tested. Then we expect 60 (that’s 5%) of them to be steroid users and the other 1,140 to be ‘clean’. Of the 60 cheats, we expect 95%, that’s 57 of them, to be correctly identified by the drug testers. But of the 1,140 clean competitors, 57 (that’s 5% of 1,140) would be incorrectly recorded as drug-free by the testers.
These are sobering statistics. The testing of 1,200 players would result in 114 positives. Of these, 57 would be guilty of drug taking and 57 would not. So, if any player tests positive there is only a 50% chance that he or she has taken drugs.
What we have described here is an example of a very important piece of reasoning about conditional probabilities, first pointed out by the Reverend Thomas Bayes of Tunbridge Wells in 1763, in an article entitled ‘Essay Towards Solving a Problem in the Doctrine of Chances’. What Bayes tells us is the relationship between the probability that an athlete is a drug taker given a failed test and the probability that a drug taker fails the test. Suppose we label by E the event that the drug test is positive, and by F the event that an athlete is taking drugs, then:
It is very important to recognise that P(E/F) and P(F/E) are different things. Prosecuting counsels in the courts are notorious for trying to bamboozle jurors into thinking they are the same, an error that is called ‘the prosecutor’s fallacy’.
What we want to know is P(F/E). In our example with the 1,200 players we know that P(F) = 0.05 and so the probability that an athlete is not using drugs is given by P(not F) = 0.95. The accuracy rate for the test is 95% and so P(E/F) = 0.95. We saw that 57 out of 1,200 (i.e. 4.75%) drug-free athletes tested positive and so P(E/notF) = 0.0475. What the Reverend Bayes showed is that all these quantities are related by a single formula:
P(F/E) = {P(E/F) × P(F)}/{P(E/F) × P(F) + P(E/not F) × P(not F)}
In our example, this becomes:
P(F/E) = {0.95 × 0.05}/{0.95 × 0.05 + 0.0475 × 0.95} = 0.513
So Bayes’ formula shows that P(F/E) is completely different to P(E/F). In our example P(F/E) is unacceptably small and far better testing would be required to discriminate more accurately between drug takers and clean athletes.
SUPPOSE THAT A football match takes place between the Reds and the Blues and the probability that the Reds score a goal is p and the probability that the Blues score a goal is 1–p. If an odd number of goals were scored what is the probability that the Reds won the game?
Well, if only one goal is scored then the probability that the Reds win the game is just p, the probability they score that single goal. But what if three goals are scored? The possible scoring sequences by the Reds (R) and Blues (B) and final results are as follows:
The probability of each of these eight results occurring is obtained by multiplying the probabilities for each goal occurring,1 so for example, the probability of RRB is (1–p) × (1–p) × p = (1–p)2p and so on.
What is the probability of the Reds winning a three-goal game? It is simply the sum of the probabilities of the four ways in which they can win the game: RRR with probability p3, plus that for the sequences RRB, RBR and BRR, each with probability p(1–p)2. The total probability that the Reds win a three-goal game is therefore:
P (Red win) = p3 + 3p(1–p)2 = p2(3–2p)
If the Reds are the stronger team and much more likely to score, with p = 2/3, say, then their chance of winning the game is P = 20/27, just slightly more than 2/3 (= 18/27). If the two teams are evenly matched and p = ½ + s, where s is a very small quantity2 then p2(3–2p) is approximately:
P (Red win) = ½ + 3s/2
If s is zero and the teams are equally likely to score then P = ½ too and they are equally likely to win the three-goal match. But if s is positive there will be a very small bias towards the Reds scoring a goal which gets amplified into a greater likelihood of 3s/2 that they will win the match. We see that they are more likely to win when three goals are scored than if they are in a game where only one goal is scored. This doesn’t mean that the Reds will win the match of course. Sometimes the weaker team does win, but the more games are played the greater the chance that the ‘better’ team will win in the long run.
THERE ARE TWO athletics events where you try to launch the body the greatest possible height above the ground: high jumping and pole-vaulting. This type of activity is not as simple as it sounds. Athletes must first use their strength and energy to launch their body weight into the air in a gravity-defying manner. If we think of a high jumper as a projectile of mass M launched vertically upwards at speed U then the height H that can be reached is given by the formula U2= 2gH, where g is the acceleration due to gravity. The energy of motion of the jumper at take-off is ½ MU2 and this will be transformed into the potential energy MgH gained by the jumper at the maximum height H. Equating the two gives U2= 2gH.
The tricky point is the quantity H – what exactly is it? It is not the height that is cleared by the jumper. Rather, it is the height through which the jumper’s centre of gravity is raised, which means it is a rather subtle thing, because it is possible for a high-jumper’s body to pass over the bar even though his centre of gravity passes under it.
When an object has a curved shape, like an L, it is possible for its centre of gravity to lie outside of the body. This possibility allows a high jumper to control where his centre of gravity lies, and the trajectory it follows when he jumps. The high-jumper’s aim is to get his body to pass cleanly over the bar whilst making his centre of gravity pass as far underneath the bar as possible. In this way he will make optimal use of his explosive take-off energy to increase H.
The simple high-jumping technique that you first learn at school, called the ‘scissors’ technique, is far from optimal. In order to clear the bar your centre of gravity, as well as your whole body, must pass over the bar. In fact your centre of gravity probably crosses about thirty centimetres above the height of the bar. This is a very inefficient way to clear a high-jump bar and techniques used by top athletes are much more elaborate. The old ‘straddle’ technique involved the jumper rolling around the bar with their chest facing it and was the favoured method of world-class jumpers until 1968, when the American Dick Fosbury amazed everyone by introducing a completely new technique – the ‘Fosbury Flop’ – which employed a backwards flop over the bar. It won him the gold medal at the 1968 Olympics in Mexico City. Fosbury’s technique was much easier to learn than the straddle and it is now used by every good high jumper. The more flexible you are the more you can curve your body around the bar and the lower will be your centre of gravity. The 2004 Olympic men’s high-jump champion Stefan Holm, from Sweden, is rather small by the standards of high jumpers but is able to curl his body to a remarkable extent, making it almost U-shaped at his highest point. He sails over a bar set at 2m 40cm but his centre of gravity goes well below it.
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