Plus Blog

September 3, 2012
Sarah Storey

Sarah Storey (photo from Paralympics GB)

Today we say a fond farewell to the velodrome after the final track cycling events on the weekend. From Sarah Storey's first gold for Great Britain on Thursday to the cat-and-mouse gold medal game between compatriots Anthony Kappes and Neil Fachie of the Individual B Sprint yesterday, it has been a fabulous show. To celebrate we calculate just how Storey and her fellow cyclists are able to reach such astonishing speeds in Gearing up for gold.

Storey also features in three maths challenges from the BBC Two Learning Zone (developed with input from us, the Millennium Mathematics Project): the Key Stage 1 - Timing and Ranking Challege, the Key Stage 2 - Timing Challenge and the Key Stage 3 - Measuring Challenge.

And though the Paralympic and Olympic races are over for now, we can look forward to many more split second finishes in this outstanding venue in the coming years. Read more about the maths behind the Velodrome's iconic curves Leaning into 2012 and How the Velodrome found its form.

August 31, 2012
Computer model of the London 2012 swimming venue

Precision is crucial. Computer model of the London 2012 Aquatics Centre. Image courtesy London 2012.

The Olympic Stadium and the Aquatics Centre are taking centre stage again as medals are being awarded in both athletics and swimming. Not any old pool or running track qualifies as an Olympic venue, partly because the accuracy of the lap length is very important. Make the track or pool too short and the distance covered will fall short of the required distance more and more as the race progresses. Any records set will be invalid when ultimately checked by laser ranging. But even when tracks and pools stay within the IAAF and FINA approved tolerances small errors can build up and make a real difference to race timings. Find out more in When errors snowball.

August 30, 2012

Wheelchair racing is one of the most exciting disciplines in the Paralympics. And it's not just a wheel-based equivalent of Olympic racing: John D. Barrow, mathematician, cosmologist and prolific popular science writer, has spotted an important difference:

Wendel Silva Soares

The Brazilian athlete Wendel Silva Soares in a 400m wheelchair race. Image: Marcello Casal.

The first international sports events for disabled athletes held in conjunction with an Olympic Games were in Rome in the summer of 1960. The term Paralympics was only coined four years later at the Tokyo Olympics. At first, wheelchair athletes used conventional heavy wheelchairs (7-18kg) and only contested events up to 200 metres. The first wheelchair-borne marathon competitor competed in the 1975 Boston Marathon. New races were organised and gradually purpose-built racing chairs appeared. By the 1980s they had become lightweight (4-10kg) and technically sophisticated. The first mile completed in under four minutes by a wheelchair athlete came in 1985 and the roster of Olympic events was considerably extended for both men and women. Intense competition has driven down records in the commonest events on the track and in the marathon, which now features in most top-flight big city races, like London and Boston.

It is very interesting to look at the trends in world record performances for able-bodied and para-athletes. They are both very well defined but quite different. Able-bodied athletes are faster up to about 400 metres but after that their average speed quickly falls behind the wheelchair performances.

The two tables below show the world record times for the Olympic running and wheelchair events, for both men and women, together with the average speed of the athlete in each case – this is just the distance covered divided by the time recorded. If you like to think of speeds in miles per hour rather than metres per second, then 10m/s corresponds to 22.4mph, so Haile Gebrsellasie cruises around the marathon course at about 12.7mph.

Table 1: Record times and average speed for able-bodied athletes.
Men's eventRecord timeSpeed m/sWomen's eventRecord timeSpeed m/s
100m9.5810.44100m10.49s9.53
200m19.19s10.42200m21.34s9.37
400m43.18s9.26400m47.60s8.40
800m1m 40.91s7.93800m1m 53.28s7.06
1500m3m 26.00s7.281500m3m 50.46s6.51
5000m12m 37.53s6.605000m14m 11.15s5.87
10000m26m 17.53s6.3410000m29m 31.78s5.64
Marathon2hr 03m 02s5.72Marathon2hr 15m 25s5.19
Table 2: Record times and average speed for wheelchair athletes.
Men's wheelchair
event
Record timeSpeed m/sWomen's wheelchair
event
Record timeSpeed m/s
100m13.76s7.27100m15.91s6.29
200m24.18s8.27200m27.52s7.27
400m45.07s8.88400m51.91s7.71
800m1m 31.12s8.77800m1m 45.19s7.61
1500m2m 54.51s8.561500m3m 21.22s7.45
5000m9m 53.05s8.435000m11m 16.96s7.39
10000m19m 50.64s8.4010000m24m 21.64s6.84
Marathon1hr 20m 14s8.77Marathon1hr 38m 29s7.14

You would expect to find that the average speed of each record run falls as the distance of the event gets longer. For able-bodied running records the trend is remarkably well defined. In the graph below you can see the systematic trends for the men’s and the women’s events. The average speed, $U$, in m/sec varies as a power of the distance, $L$, in metres. In other words, $U$ is approximately equal to $aL^ b$ for some constants $a$ and $b.$ Such a relationship is called a power law. The constant $a$ is not that important here as it is just a scaling factor. The interesting quantity is the exponent $b.$ For men this is $-0.109$ and for women it is $-0.111.$ So $U$ is proportional to $L^{-0.109}$ for men and to $L^{-0.111}$ for women.

Records and average speeds

Left: average speed versus distance for men. Right: average speed versus distance for women. The relationship between average speed and distance follows a power law.

The corresponding results for the average speed trends in the wheelchair events are strikingly different. If we don’t pay too much attention to the 100m and 200m events because of the strong effect that getting started and building up speed has on the total time, we can see already from the table of average speeds that there is hardly any fall off of average speed with distance. The athlete’s technique reaches the fastest turnover rate quite quickly and can maintain it over very long distances. This is reflected in the exponents in the power law being close to zero, so $U$ nearly remains constant as the distance $L$ varies. The exponent is $-0.005$ for men and to $-0.021$ for women.

Records and average speeds

For the wheelchair events the exponents in the power law are close to zero.

We can also see from the tables that the average speed achieved during the marathon record is higher than in the shorter distances like the 10000m, and even the 5000m. There are several reasons that contribute to this anomaly. The shorter distance records are set on the track and require the negotiation of two bends on each 400m circuit of the track. Able-bodied runners are not much affected by this but it more problematic for wheelchairs and also harder for them to overtake. This slows the wheelchair on the bends compared with the two straights. The marathon is raced over flat road courses with a minimum of twists and turns and is much better for wheelchair racing although in practice the courses are optimised for the able-bodied runners. Also, wheelchair marathons are more competitive and frequent than 10000m track races and the records are under far more pressure with larger fields of competitive entrants. The tactical nature of 10000m track races also often leads to slower times than in road races.

The unusually small variation of speed with distance also allows wheelchair athletes to be competitive over a far greater range of distances than able-bodied athletes. David Weir has won the London marathon wheelchair race on four occasions but also has Olympic medals for the 100m, 200m, 400m, 800m, 1500m and 5000m wheelchair events. No able-bodied athlete could ever hope to succeed in more than three of those events. Our average speed analysis shows why it is a possibility for para-athletes.

The fact that wheelchair racers have very similar speeds over the whole range of competitive distances from the sprints to the marathon opens up an exciting possibility for wheelchair athletics. We saw recently that some athletics promoters were very keen on the idea of Usain Bolt racing the new 800m world record holder David Rudisha over the intermediate distance of 400m. In fact, this would not be as interesting as it sounds because Bolt would win very easily, although if the distance was increased to 500m it could get interesting.

Such match ups don't work well for able-bodied athletes because of the significant differences in speed from one event to the ones on either side of it in distance. However, in wheelchair racing it would be possible to have a super championship featuring the champions and record holders from all the competitive distances over one (or two) intermediate distances.

A version of this article has appeared in Barrow's book 100 Essential Things You Didn't Know You Didn't Know About Sport.

You can buy the book and help Plus at the same time by clicking on the link on the left to purchase from amazon.co.uk, and the link to the right to purchase from amazon.com. Plus will earn a small commission from your purchase.
August 29, 2012
Oscar Pistorius

Oscar Pistorius. Image: Elvar Pálsson.

If the Olympics weren't enough for you, then you're in for another eleven days of top-performance sport: the Paralympic Games will open tonight. The first events for disabled athletes were held in conjunction with Olympic Games in Rome in the summer of 1960, although the term Paralympics was not used until the 1964 Tokyo Olympics. Since then the athletes, as well as their equipment, have progressed enormously. While the first wheelchair athletes used conventional wheelchairs, these days competitors race along in sophisticated lightweight racers. And advances in prosthetic limbs are even more impressive, as "blade runner" Oscar Pistorius exemplifies.

This got us thinking about the role of engineering and technology in sport. New technologies mean that the performance of athletes and their equipment, whether Paralympic or Olympic, can be measured, tested and simulated in ways that previous generations could only dream of. Tiny winning margins mean that any advantage that can be gained in this way is worth the effort. Last year we spoke to several researchers in sport technology about how their work makes a difference to sport: from designing sophisticated bobsleds for the Winter Olympics to the newly emerging "science of coaching". You can read about this in our article Making gold for 2012 or listen to the interviews in our podcast.

If you're near London, then you might want to pop along to the Royal Academy of Engineering, which is staging some interesting events for the Paralympics. From September 3rd to 14th it is hosting the Rio Tinto Sports Innovation Challenge Exhibition. The specially selected exhibits illustrate new sporting opportunities ranging from equipment through to radical new sporting events and competition models to facilitate active lifestyles for people with disabilities. On September 6th there's a lecture by Amit Goffer called Powered exoskeletons: Overcoming vertical mobility impairments. Goffer is the inventor of the first commercially available upright walking technology, enabling wheelchair users with lower-limb disabilities to stand, walk, and even climb stairs. And on September the 13th there's the Meet the Athletes drinks reception. It's an opportunity to meet and greet athletes that have taken part in disability sports, including an exclusive viewing of the specialist sports equipment they have used. To find out more about these events, click on the links or visit the Royal Academy of Engineering website.

August 24, 2012

The British Science Festival comes to Aberdeen this year from 4th to 9th September. It's the largest annual public science event in Europe. This year's theme is Energising minds — and there's plenty of maths on offer to energise yours. The maths events in this year's main programme are:

seal

Stats explores the interaction between seals and cod.

  • Fishy figures — How do statistical methods improve our understanding of the sea and how we use it? How do these methods improve our knowledge of the numbers of whales, the protection of humans from shellfish toxins, and the interaction between cod and seals? Find out how statistics are used to understand the world around us.
    Wednesday 5 September, 13:00–15:00
    Audience level: everyone
    Price: free
    Book here

  • Expanding minds and universes: Presidential Lecture by John D. Barrow — Join John D. Barrow to hear about how Einstein made it possible for cosmologists to study whole universes. Find out how expanding universes, rotating universes, chaotic universes, inflationary universes, accelerating universes and multiverses were all discovered. What is the current best description of the Universe and what problems still remain to be solved by mathematical physicists? Followed by a wine reception 16:30–17:30 sponsored by the Edinburgh Mathematical Society.
    Friday 7 September 15:30–16:30
    Audience level: all adults
    Price: free
    Book here

  • In tune with mathematics — What does it mean to be musically 'in tune'? The story takes us from Pythagoras to the modern day. We'll see why the standard Western tuning of instruments is in fact out of tune and we'll see how mathematics revolutionised the music industry via software that helps singers sing in tune. Along the way we hope to solve the puzzle of the Beatles' Magical Mystery Chord.
    Saturday 8 September 15:00–17:00
    Audience level: everyone
    Price: free
    Book here

  • Turing: the human vs the machine — Alan Turing, one of the truly original thinkers of the last century, was born in 1912. We celebrate his centenary with an overview of his achievements in logic, mathematics, computing and artificial intelligence. The Turing test asks whether you can distinguish, blindfolded, a human's conversation from a computer's. Join in and cast your vote.
    Sunday 9 September 15:30–17:30
    Audience level: everyone
    Price: free
    Book here

  • The maths and computing magic show — Witness some amazing magic tricks and sneak behind the scenes to explore the maths and computing secrets behind them. Mathematics and computer science are behind today's technological wizardry, and help us understand our own brains! Peter McOwan and Matt Parker are both scientists and magicians, and they will be your guides to the secret world where science and conjuring meet.
    Sunday 9 September 18:00–20:00
    Audience level: families
    Price: £5
    Book here

And there's much more interesting stuff besides. To find out, have a look at the programme.

August 14, 2012

Never afraid of a challenge, before the start of the London 2012 Games we issued predictions for the total medal count for the top 20 countries. They were based on a mathematical model that took account of a country's GDP and population, its performance in 2008 and the home advantage bestowed on Great Britain and also China, who hosted the Games in 2008 (see Mapping the medals).

So how did we do? The first thing to notice is that, despite team GB's gold rush, it actually performed worse in terms of total medals won that we had predicted, ending up in fourth place rather than in second — despite the home advantage, the likes of Russia and China are hard to beat. Hungary makes a surprising appearance, entering in 14th place while we didn't have it down as making the top twenty at all. New Zealand came in 18th, unpredicted by us, and Iran also just about squeezed in, tying with Jamaica in 20th place. And Kenya, which we had down to come 18th, narrowly missed the top twenty, coming in as 21st. Other than that all but three countries from our original list stayed within three ranks of our predictions. Japan and the Netherlands both performed better than we had predicted, Japan coming in as 6th rather than 11th and the Netherlands at 11th rather than 16th. Belarus gave a disappointing performance compared to our mathematical benchmark, coming in as 16th rather than 11th. Overall, 19 of our top 20 predicted countries finished in the top 20 (top 22 actually as 3 countries were tied on 20th).

2012 Predicted
Position
2012 Predicted
Medals
United States1112
Great Britain279
Russia377
China476
Australia553
France 642
Germany642
South Korea832
Ukraine929
Italy1028
Japan1125
Cuba1225
Belarus1321
Canada1419
Spain1419
Netherlands1617
Brazil1716
Kenya1815
Kazakhstan1815
Jamaica2012
2012 Actual
Position
2012 Actual
Medals
United States1104
China288
Russia382
Great Britain465
Germany544
Japan 638
Australia635
France834
South Korea928
Italy928
Netherlands1120
Ukraine1120
Canada1318
Hungary1417
Brazil1417
Spain1417
Cuba1514
Kazakhstan1813
New Zealand1813
Jamaica2012
Belarus2012
Iran2012

Where could we improve?

We failed to note that country populations and GDP nearly always rise between Olympics, whilst the number of medals available in 2008 and 2012 were roughly the same.

This means that to use the equation relating population and GDP to medal count derived from 2008 we should have scaled the 2012 country data relative to the whole world's population and GDP. Imagine your population rose between 2008 and 2012 — you might think that you would have a greater chance of winning a medal. But if the whole world's population grew, then your chances wouldn't have increased. We need to use population (and GDP) scaled to the whole world.

This makes little difference to our predicted ranking, except for moving Brazil up from 17th to 16th place and Kazakhstan down from 18th to 19th, and acts to slightly dampen our medal counts. These new counts are shown below against the real 2012 results:

Revised Predicted
Position
Revised Predicted
Medals
United States1106
Great Britain275
Russia373
China472
Australia550
France 640
Germany640
South Korea831
Ukraine928
Italy1027
Japan1124
Cuba1223
Belarus1320
Canada1418
Spain1418
Netherlands1616
Brazil1616
Kenya1815
Kazakhstan1914
Jamaica2011
2012 Actual
Position
2012 Actual
Medals
United States1104
China288
Russia382
Great Britain465
Germany544
Japan 638
Australia635
France834
South Korea928
Italy928
Netherlands1120
Ukraine1120
Canada1318
Hungary1417
Brazil1417
Spain1417
Cuba1514
Kazakhstan1813
New Zealand1813
Jamaica2012
Belarus2012
Iran2012

External links

The BBC's More or Less team conducted a similar statistical prediction of what countries might be expected to win in the way of medals with the help of the Kellogg School of Management at Northwestern University; here is their analysis of the results.

The Guardian teamed up with the Royal Statistical Society and four academics at Imperial College London to produce very interesting alternative medal tables for the 2012 Games, taking into account factors including GDP, population and team size. You can see their final alternative medal table winners here.

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