Image

The Paralympics

The Paralympic Games include athletes with mobility impairments, visual impairments, brain damage and other disabilities.  They started in the UK in 1948 with a small group of men who had been injured in the Second World War.  The first post-war Olympic Games took place in London that year, and on the opening day Stoke Mandeville Hospital, which has a world-famous spinal injuries centre, organised a sports competition for some of the patients.  The first official Paralympic Games, no longer restricted to war veterans, were held in Rome in 1960.  The Paralympic Games now operate in parallel with both the summer and winter Olympics - and are called the 'Paralympic' Games from the Greek word 'para' meaning beside or alongside.

Team events and some individual events are for specific categories of athletes, such as wheelchair basketball, goalball and judo for the visually impaired, boccia for people with cerebral palsy (brain injuries), and rowing for people who only have use of their arms.  In these events, people with other disabilities are excluded, and any variations in disability within the category are ignored.

In most individual events, athletes are divided into six broad categories - amputees, people with cerebral palsy, people with intellectual disabilities, wheelchair users, visually impaired people, and Les Autres.  'Les Autres' is the French term for 'others', and includes people with dwarfism, multiple sclerosis and congenital deformities.  Within each of these categories, athletes are then further sub-divided into classes according to the degree to which their disability impacts on their performance.

Classifying athletes

It isn't only Paralympic sports where people are classified in some way.  Golfers have their handicap as do race horses.  Boxers and weight-lifters are classified according to their body weight, and many amateur sports classify people on the basis of what they have achieved so far. 

Ideally each class should be as homogeneous as possible but different from other classes on the basis of certain criteria.  In the case of the Paralympics, the criteria include sex, the nature of the event, the disability category and the degree to which the athlete's individual level of disability affects their performance.  The classification needs to be seen to be fair, as far as possible.  The Paralympics are a high stake competition, so the process also needs to be rigorous. 

There are, however, many problems in grouping athletes into classes.  It is difficult to apply handicapping in a consistent way across different classes.  Racing conditions and changes in technology mean that classification frequently has to be revised.  Whether a class is small or large may give athletes an unfair advantage or disadvantage.  There have also been instances of cheating, with athletes claiming to be more disabled than they actually are in the hopes of being put into a class in which they would have an unfair advantage.

One way of avoiding such problems is to use a mathematical algorithm which focuses on athletes' performances, rather than the degree of their disability.  This also avoids the need for committees to meet to debate which class an athlete should be put into and how classification should be updated as equipment or other factors change.

One proposed method involves calculating an overall median for an event, and also calculating median performances for each class within that event.  The ratio of the median for a class compared to the overall median for the event tells us how good a particular class is relative to the others in the event.  Competitors' performances are then scaled using this ratio to get their score.  This process is illustrated at the end of this article. 

Setting students

The classification of athletes for the Paralympics is a very similar process to that of classifying students for mathematics and other lessons.  We might use criteria such as exam performance yet we are aware that this is a 'one off' snapshot of performance and therefore may not be entirely representative of a student's ability. To try and get a broader picture of performance we often add teacher assessment. Our attention may then turn to student confidence as we try and aim for conditions that will maximise a particular student's performance. Does the discussion of classification for the Paralympics above help further refine or inform our setting of students?

In the classroom - the school sports days

So what is the relevance here to school sports days and helping students to develop their mathematical thinking? The challenges we face when planning a sports day include:

• what events to choose
• how to allocate students to each event
• whether we need some sort of handicapping system or whether it is simply a 'flat playing field'?

The first challenge is taken up in one of this month's stage 1 problems, Our Sports. This encourages students to look at what events they might want to include in their sports day. This could be developed at stages 2 and 3, as a sampling problem.  For instance, how many students should be asked to get a fair impression of the whole school view?  Does it matter how good at sport those asked are? What would a representative sample look like?

The second challenge depends on the purpose of the sports day.  For the Paralympics it is about peak performance and winning your event to gain the coveted gold medal. It is entirely competitive.  For a school sports day it could be that everyone should take part, which raises the question of whether  the most athletic should take all the 'medals' or not.  There may be other issues which are as important as individual performances.  The purpose will inform how students are allocated to each event.  How important is it that we are modeling our school  sports day on the real world of sport?

Students could be allocated to a particular event according to:

• previous performance
• their preferences
• ensuring everyone takes part in one event
• ensuring events are of equal size 
• ??

By choosing one event - such as the 100m sprint - students could investigate the impact of different classification strategies.  They could select participants using the their own classification strategies, try out the race with the different groups and then compare outcomes. 

But how will they represent the results in order to make meaningful comparisons?  For stage 2 and 3 students this could lead to the idea of an average as in some way typifying a particular group, with the range giving an indication of the variability within that group.  Stage 3 and 4 students could think further about why the median might be preferable to the arithmetic mean.

The third challenge is informed by what we mean by 'fair'.  This could lead to an investigation of different handicapping systems, such as that for golf or horse-racing.  Students could explore how the system works and what it could be like in the school sports day.  In school we may want the events to be fair yet perhaps not go as far as a handicapping system.

Questions to consider

How might the size of class affect your chances of winning?

• In a school setting, how might the size of the set/group affect a student's attainment outcomes?
• On what grounds do we/might we make a correlation between group size and ability?

What advantages are there in using median performances rather than mean, minimum or maximum performances for each class?

• How does this apply in a school setting in the way we may choose to group students?

Stage 3 students could also look at How Would You Score It?

Example of classification using median performances

Thanks to Professor David Percy, University of Salford, for his help with this section.

Here is a set of 50 random scores, between 0 and 100:

Suppose we want to divide the 'athletes', whose scores these are, into three equivalent groups so that we can judge who the overall medal winners should be.  We could base our classification on the median, so that we make the median of each group the same, then having scaled all the scores, see whose score is best. 

This process is illustrated below.

I start by subdividing the scores into three classes, according to whether the score is in the range 0 - 33 (Class A), 34 - 66 (Class B), and 67 - 100 (Class C).  If a high score is best, this might indicate people who are severely affected by their impairment, those moderately affected, and those only slightly affected.

Class A:   1, 5, 29, 23, 25, 10, 10, 28, 17, 23, 17, 4, 32, 2, 26, 17, 20, 6   (18 members)

Class B:   66, 44, 60, 53, 50, 55, 44, 64, 51, 54, 38, 53, 42, 62   (14 members)

Class C:   92, 76, 76, 84, 77, 80, 92, 67, 72, 79, 74, 79, 95, 68, 78, 80, 74, 83   (18 members)

The overall median for the 50 scores is 53, and the three class medians are:

m_A = 17,   m_B = 53,   m_C = 78.5

You could ask the question here whether the median of the central group is necessarily the same as the overall median (or is it simply a consequence of the members of the groups being symmetrically distributed?).

A simple scaling factor for each group can be calculated by dividing the minimum median by each class median, giving:

f_A = 1,   f_B = 0.321,   f_C = 0.217

(17 x 1 = 17, 53 x 0.321 = 17, 78.5 x 17 = 17)

The raw scores are then scaled by these factors to give adjusted scores from which medal winners can be determined.  

It turns out that all the medal winners would be in Class A, if a high score was best.  These are the adjusted scores:

Class A:   1, 5, 29, 23, 25, 10, 10, 28, 17, 23, 17, 4, 32, 2, 26, 17, 20, 6   

Class B:   21.2, 14.1, 19.3, 17.0, 16.0, 17.6, 14.1, 20.5, 16.4, 17.3, 12.3, 17.0, 13.5, 19.9

Class C:   19.9, 16.5, 16.5, 18.2, 16.7, 17.3, 19.9, 14.5, 15.6, 17.1, 16.0, 17.1, 20.6, 14.7, 16.9, 17.3, 16.0, 18.0  

The ranges of classes with higher medians are reduced by the scaling process.  Whereas for Class A 0 scales to 0, and 33 scales to 33, for Class B the minimum (34) and maximum (66) scale to 10.9 and 21.2 respectively, and for the minimum (67) and maximum (100) scale to 14.5 and 20 respectively.  This favours the class with the lowest median, as we have seen.

This results from choosing data by randomly selecting integers from a range, which implies that, eg. 50 is twice as good as 25.  Sporting data is not random and minimum scores are unlikely to be close to the bottom of the class, so such extreme results as this would be much less likely.

What difference would it make if class maxima were used, instead of class medians?  If we assume that a high score is good, might this make for a fairer system of score adjustment?

I once made a school visit in just about the heaviest rain I've ever experienced. It came down with such force that it made you wet twice, once on the way down and again as it bounced up to waist height. It continued just as hard throughout the first half of the morning, and as morning break neared a deputation of worried-looking Y6 girls approached the teacher. "Please, miss, you're not going to let a few drops of rain stop our rugby practice, are you?"

 

Image

As the Olympics get closer and closer it seems to me that one of the healthiest developments in recent years has been the increased opportunities for girls to participate in sport. I've recently been doing One-To-One tuition in the middle school I taught at 25 years ago. Of course the new buildings and the computer rooms are a fascinating contrast to the days when our computer was a Sinclair ZX81, but in 1985 a girls' football team was just as much pie-in-the-sky as a computer suite. The 2011 version is that we had to modify the timetable because one of my pupils wasn't prepared to miss her football practice. 

 

Going back even further in my own experience, I spent the first couple of years in secondary school as a boarder, and we'd spend hours simulating our own Test Matches. All you needed was a pencil and an exercise book, and if they'd ever thought to look, my teachers would have been amazed by that exercise book. To call my exercise books in mathematics untidy would be to flatter them generously - crossings out, ink blots, wrong answers and very few correct ones.

What a contrast there was in the appearance of my Test Match book - each game would take several days to play and there'd be perhaps 3000 separate pieces of information all impeccably and accurately recorded. On reflection there's no great insight needed to explain why I might have been at the bottom of the class in mathematics lessons and yet able to call upon all the mathematics I needed to explore sporting situations - I spent two weeks in the sick bay and was perfectly happy that my only company was an old copy of Wisden Cricketers' Almanack.

A good stimulus can do much more than simply encourage children to work with a bit more enthusiasm. It can unlock levels of performance way beyond all expectations. I joined a Y6 teacher using a computer package about Grand Prix racing called "Cars - Maths In Motion". Now to my mind Formula One has not one single attribute that qualifies it as a sport, but I don't mind admitting that motor racing does nevertheless stimulate some fine games. The package wasn't about cars whizzing around; indeed, there was no graphical component at all. Instead, children had to prepare a car for a race, making the best compromise between speed, power, brakes, etc, and to do this they had to do a vast amount of work involving estimating, decimals, and percentages.

The work lasted for several lessons, and the teacher told me that at one stage a visitor joined the class. She was unconvinced by the teacher's claims that a computer game about motor-racing was contributing to the pupils' mathematics and their understanding of percentages. The visitor buttonholed a child at random, choosing someone who was by no means a star in the class.

"What's 92% of 228?"

The child looked puzzled, and the teacher's heart sank.

"Go on Claire", she said, "I know you can do it. What's the answer?"

"About 209, I suppose", muttered Claire nervously.

When prompted, Claire explained her reasoning (all of which she had of course done in her head):

"10% of 228 is about 23, so if I do 228-23 I'll have 90%, but to get 92% I'll have to add back on another 2%. 1% is about 2, so the answer is about 228-23+2+2 = 209".

The visitor sat down to do some calculations on a piece of paper and several minutes later came up with a similar answer (the exact value is 209.76). When the visitor - somewhat more satisfied - had left, the teacher asked Claire why she hadn't given the answer immediately.

"It seemed too easy", she said, "I thought there must be a catch"!

This was, incidentally, another occasion when girls surpassed my rather bigoted expectations. Not only did two girls win the race, they overtook a further pair of girls whose car ran out of fuel on the final lap, and another girl left her sickbed not to miss the race. There are few computer packages which have been unsurpassed for more than twenty years, and you can find out more about this one at Maths In Motion Challenge for Schools.

 

Image

A little while ago I visited Arsenal Football Club's Double Club initiative. Arsenal-sponsored teachers are linked with local schools and teach lessons in basic subjects. The lessons have football-linked themes and the Arsenal teachers offer additional football lessons. I'm as keen on football as anyone, but even I could recognise there might be some downsides, so I visited with at least a small amount of scepticism in my mind. I certainly recognised benefits - the pupils, many of whom had histories of little academic success, were enthusiastic and had a real commitment to the work. The DCSF evaluation (ref DCSF-RR101, ISBN 978 1 84775 422 6) was carried out by NFER, and reports:

 

Whether or not you want to build your teaching around football there seems a clear message here. Pupils who might not normally be motivated towards basic skills work were prepared to work hard because the work was presented in a way that they could relate to.

 

Image

Yes, I do think the Cars - Maths In Motion program is a terrific stimulus, but I'm equally sure that perfectly ordinary ideas also can have significant effects. Back at my middle school of a generation ago the sports staff were so committed to athletics that I was convinced that they taught the children far more about decimals and measurement than the entire mathematics team. Equally, one lesson I drew from my Double Club visit was that the material itself didn't have to be particularly creative or exciting - simply presenting a standard task adapted to a football context ensured that the routine task was perceived as having a relevance and a meaning that's so often been lacking in maths lessons. Mathematics teaching is notorious for being totally divorced from anything real, but because sport is of interest to so many it may offer a motivating context for the most prosaic of activities.

There are 2 well-known types of tournament formats that sport schedulers use.

• Single Elimination Tournaments.
• Round Robin Tournaments.

Single Elimination Tournaments.

In this format, one defeat is enough to eliminate a team from the tournament.

Scheduling a single elimination tournament is relatively easy. The first step is to get the number of teams to a power of two: 2, 4, 8, 16, 32...

The schedule for 8 teams is shown below.

Image

With 20 teams, you might select eight of those teams to have a preliminary round. The four winners would then join the other twelve to fill out the sixteen-team field.

If you have 24 teams, you could have everyone compete in a preliminary round; this would leave 12 teams. Then, 4 of those teams (randomly selected) could get a bye and the other 8 teams would play to find the other 4 contestants for the next round.

Round Robin Tournaments.

In a Round Robin tournament every team plays every other team .

There is a systematic approach to scheduling a Round Robin tournament. This method assumes that there are enough fields / pitches / courts so that all the games in a round can be played simultaneously. The technique is called the polygon method .

Round Robin scheduling: Even number of teams.

Let N = number of teams in the tournament. There will be N -1 rounds (each team will play N-1 games). Since each team will play every other team once, no team will be idle during any of the rounds.

Let us schedule a round-robin tournament for 8 teams numbered from 1 to 8.

Draw a regular (N -1) sided polygon (i.e., a heptagon for 8 teams). Each vertex and the centre point represents one team.

Image

Draw horizontal stripes as shown below. Then, join the vertex that has been left out to the centre. Each segment represents teams playing each other in the first round.

Image

So (7, 6), (1, 5), (2, 4) and (3, 8) play in the first round.

Rotate the polygon 1/(N-1)th of a circle (i.e. one vertex point). The new segments represent the pairings for round two.

Image

So (6, 5), (7, 4), (1, 3) and (2, 8) play the second round.

Continue rotating the polygon until it returns to its original position.

Image

One more rotation will bring the polygon back to its original position.

If A, B, C and D are the fields / pitches / courts, the schedule could look like this:

We can also rotate the teams around so that each team plays in every field / pitch / court at least once (at present team 8 always plays in D).

Round Robin scheduling: Odd number of teams.

Let N = number of teams in the tournament. There will be N rounds (since each team will play every other team once, and will be idle for exactly one round ).

Let us work out the schedule for 7 teams, numbering the teams from 1 to 7. Draw a regular N-gon (heptagon for 7 teams). Each vertex represents one team.

Image

Draw horizontal stripes as shown below. The vertex that has been left out gives the idle team. Each segment represents teams playing each other in the first round.

Image

So (7, 6), (1, 5) and (2, 4) play in the first round.

Rotate the polygon 1/Nth of a circle (i.e. one vertex point.) The new segments represent the pairings for round two.

Image

Continue rotating the polygon until it returns to its original position.

Image

One more rotation will bring the polygon back to its original position. Therefore, the schedule could look like this:

Why does this work?

The restriction that no vertex has more than one segment drawn to/from it ensures that no team is scheduled for more than one game in each round.

Restricting ourselves to horizontal stripes ensures that no segment is a rotation or reflection of another segment. This means that no pairing will be repeated in a future round.

Notice that in the case where N (no. of teams) was odd, by having only one idle team in each round, the tournament can be completed in the minimum number of rounds.

Arunachalam Y. is a member of the HeyMath! team

The decathlon consists of ten track and field events spread over two days. It is the most physically demanding event for athletes. On day one, the 100m, long jump, shot putt, high jump and 400m are contested. On day two, the competitors face the 110m hurdles, discus, pole vault, javelin and, finally, the 1500m. In order to combine the results of these very different events - some give times and some give distances - a points system has been developed. Each performance is awarded a predetermined number of points according to a set of performance tables. These are added, event by event, and the winner is the athlete with the highest points total after ten events. The women's heptathlon works in exactly the same way but with three fewer events (100m hurdles, high jump, shot, 200m, long jump, javelin and 800m).

Image

Originally, the points tables were set up so that (approximately) 1000 points would be scored by the world record for each event at the time. But records move on and now, for example, Usain Bolt's world 100m record of 9.58s would score him 1202 decathlon points whereas the fastest 100m ever run in a decathlon is 'only' 10.22s for a points score of 1042. The current world record that would score the highest of all in a decathlon is Jürgen Schult's discus record of 74.08m, which accumulates 1383 points.

All of this suggests some important questions that bring mathematics into play. What would happen if the points tables were changed? What events repay your training investment with the greatest points payoff? And what sort of athlete is going to do best in the decathlon - a runner, a thrower or a jumper?

The decathlon events fall into two categories: running events where the aim is to record the least possible time and throwing or jumping events where the aim is to record the greatest possible distance. The simplest way of scoring this would be record all the throws and jumps distances in metres, multiply them together and then multiply all the running times in seconds together and divide the product of the throws and jumps by the product of the running times, T. The Special Total that results will have units of $(\mathrm{length})^6 \div (\mathrm{time})^4 = \mathrm{m}^6/\rm{s}^4$ and spelt out in full it looks like this:

$$\mbox{Special Total, ST} = \frac{\mathrm{LJ} \times \mathrm{HJ} \times \mathrm{PV} \times \mathrm{JT} \times \mathrm{DT} \times \mathrm{SP}}{T(100\mathrm{m}) \times T(400\mathrm{m}) \times T(110\mathrm{mH}) \times T(1500\mathrm{m})}$$

 

If we take the two best ever decathlon performances by Šebrle (9026 pts) and Dvořák (8994 pts) and work out the Special Totals for the 10 performances they each produced then we get

Šebrle (9026 pts):  ST = 2.29

Dvořák (8994 pts): ST = 2.40

Interestingly, we see that the second best performance by Dvořák becomes the best using this new scoring system.

In fact, our new scoring system is not a good one. It contains some biases. Since the distances attained and the times recorded are different for the various events you can make a bigger change to the ST score for the same effort. An improvement in the 100m by from 10.6s to 10.5s requires considerable improvement but you don't get much of a reward for it in the ST score. By contrast reducing a slow 1500m run by 10 seconds has a big impact. The events with the room for larger changes have bigger effects on the total. The actual scoring points tables that are used incorporate far more information about comparable athletic performances than the simple ST formula we have invented.

Image

The points awarded (decimals are rounded to the nearest whole number to avoid fractional points) in each track event - where you want to give higher points for shorter times (T) are given by the formula

$$\mbox{Track event points} = \rm{A} \times (\rm{B} - \rm{T})^\rm{C},$$

where T is the time recorded by the athlete in a track event and A, B and C are numbers chosen for each event so as to calibrate the points awarded in an equitable way. The quantity B gives the cut-off time at and above which you will score zero points and T is always less than B in practice -- unless someone falls over and crawls to the finish! For the jumps and throws - where you want to give more points for greater distances (D) - the points formula for each event is

$$\mbox{Field event points} = \rm{A} \times (\rm{D} - \rm{B})^\rm{C}$$

The three numbers A, B and C are chosen differently for each of the ten events and are shown in this table. You score zero points for a distance equal to or less than B. The distances here are all in metres and the times in seconds.

Most importantly, the points achieved for each of the 10 events are then added together to give the total score. In our experimental ST scoring scheme above they were multiplied together. You could have added all the distance and all the times before dividing one total by the other though.

In order to get a feel for which events are 'easiest' to score in, take a look at this table which shows what you would have to do to score 900 points in each event for an Olympic-winning 9000-point total.

There is an interesting pattern in the decathlon formulae that change the distances and times achieved into points. The power index C is approximately 1.8 for the running events (1.9 for the hurdles), close to 1.4 for the jumps and pole vault and close to 1.1 for the throws. The fact that C > 1 indicates that the points scoring system is a 'progressive' one, curving upwards in a concave way; that is, it gets harder to score points as your performance gets better. This is realistic. We know that as you get more expert at your event it gets harder to make the same improvement but beginners can easily make large gains. The opposite type of ('regressive') points system would have C < 1, curving increasingly less, while a 'neutral'; one would have C = 1 and be a straight line. We can see that the IAAF tables are very progressive for the running events, fairly progressive for the jumps and vault, but almost neutral for the throws.

In order to get a feel for how the total points scored is divided across events, the Figure below shows the division between the ten events for the averages of the all-time top 100 best ever men's decathlon performances.

Image

It is clear that there has been a significant bias towards gathering points in the long jump, hurdles and sprints (100m and 400m). Performances in these events are all highly correlated with flat-out sprinting speed. Conversely, the 1500m and three throwing events are well behind the other disciplines in points scoring. If you want to coach a successful decathlete, start with a big strong sprint hurdler and build up strength and technical ability for the throws later. No decathletes bother much with 1500m preparation and rely on general distance running training.

Image

Stop Press! Kevin Mayer broke the decathlon world record at the Decastar meeting in Talence on 16th September  2018, scoring 9126 at the final IAAF Combined Events Challenge fixture of 2018.

$^i$ See http://www.iaaf.org/mm/Document/Competitions/TechnicalArea/ScoringTables_CE_744.pdf

$^{ii}$ www.iaaf.org

If you are training to be good at any sport then you are in the business of optimisation - doing all you can to enhance anything that will make you do better and minimise any faults that hinder your performance. This is one of the areas of sports science that relies on the insights that are possible by applying a little bit of mathematics. Here we are going to think about two athletics events where you try to launch the body over the greatest possible height above the ground: high jumping and pole vaulting.

This type of event is not as simple as it sounds. Athletes must first use their strength and energy to launch their body weight into the air. 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 $$U^{2}=2g H$$ where $g$ is the acceleration due to gravity. Alternatively we can think in terms of energy conservation. The kinetic energy of the jumper at take-off is $\frac{1}{2}M U^{2}$ and this will be transformed into the potential energy $M g H$ gained by the jumper at the maximum height $H$ when he is instantaneously at rest at the highest point. Equating the two gives $ U^{2}=2g H$ again.

All this sounds straightforward but 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 that the jumper's centre of gravity is raised, and that is rather a subtle thing because it makes it possible for a high jumper's body to pass over the bar even though his centre of gravity passes under the bar. When an object has a bendy shape it is possible for its centre of gravity to lie outside of the body. One way to locate the centre of gravity of an object is to hang it up from any point on the object and drop a weighted string from the same point, marking where the string drops. Then repeat this by hanging the object up from another point. Draw a second line where the hanging string now falls. The centre of gravity is where the lines of the two strings cross. If the object is a square then the centre of gravity will lie at the geometrical centre but if it is L-shaped or U-shaped the centre of gravity will not lie inside the boundary of the body at all.

Image

It is this possibility that allows a high jumper to control where his centre of gravity lies and what trajectory it follows when he jumps. The aim of our high-jumper is to get his body to pass 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.

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 goes close to 30 centimetres higher than the height of the bar. This is a very inefficient way to clear a high-jump bar. The high-jumping techniques used by top athletes are much more elaborate. The old 'straddle' technique involved the jumper rolling around the bar with their chest always facing the bar. This was the favoured technique of world-class jumpers up until 1968 when the American Dick Fosbury amazed everyone by introducing a completely new technique which involved a backwards flop over the bar and won him the Gold Medal at the 1968 Olympics in Mexico City. This method was only safe when inflatable landing areas became available. Fosbury's technique was much easier for high jumpers to learn than the straddle and it is now used by every serious high jumper. It enables a high jumper to send their centre of gravity well below the bar even though their body curls over and around it. The more flexible you are the more you can curve your body around the bar and the loweryour centre of gravity will be. 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. His body is very U-shaped at his highest point. He sails over 2m 37 cm but his centre of gravity goes well below the bar.

Image

The photographer Peter Kjelleras captures the Olympic high-jump champion Stefan Holm jumping at the World Athletics Championships in Paris, in 2003. Holm dramatically demonstrates his ability to send his centre of gravity far below the bar he is clearing 2.32 metres above the ground.

Image

Let's see if we can get a rough estimate of how well we might expect them to do. Suppose they manage to transfer all their horizontal running kinetic energy of $\frac{1}{2}MU^{2}$ into vertical potential energy of $MgH$ then they will raise their centre of mass a height of:

$$ H=\frac{U^{2}}{2g}$$

If the Olympic champion can reach $9\mathrm{\ ms^{-1}}$ launch speed then since the acceleration due to gravity is $g=10\mathrm{\ ms^{-2}}$ we expect him to be able to raise his centre of gravity height of $H=4$ metres. If he started with his centre of gravity about $1.5$ metres above the ground and made it pass $0.5$ metres below the bar then he would be expected to clear a bar height of $1.5+4+0.5=6$ metres. In fact, the American champion Tim Mack won the Athens Olympic Gold medal with a vault of $5.95$ metres (or $19^{\prime }6\frac{1}{4}"$ in feet and inches) and had three very close failures at $6$ metres, knowing he had already won the Gold Medal, so our very simple estimates turn out to be surprisingly accurate.

John D. Barrow is Professor of Mathematical Sciences and Director of the Millennium Mathematics Project at Cambridge University.