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Mathematics and Sports
John D. Barrow
Sport is just experimental mathematics. Whether we are looking at the swing and spin of cricket balls, the pirouettes of ice skaters, or the leaps of high jumpers, all are movements that can be understood by the application of simple mathematics. In these video conferences we are going to take a look at a few events in track and field athletics, football, and rowing in order to understand
better what is going on in these sports and show how very simple mathematics can shed light on what is happening in sports events.
For the applied mathematician there are two parts to applying mathematics to sports: the first is formulating the problem in a simple way and the second is solving it. The first of these steps is an example of what mathematicians call 'mathematical modelling' - taking a complicated real world problem and reducing it to its essential ingredients so that simple equations can be used to study it. We
need to do this because in practice any attempt to describe a real world problem, like a swerving tennis ball, exactly is impossible. It is too complicated. We need to know what things we can safely neglect at first - for example we might assume that our tennis ball is perfectly spherical and neglect air resistance to its motion. Then we can gradually introduce more of these details as we
progress. In fact, mathematics is most successful in telling us how the world works in precisely those areas where this method of idealization followed by successive improvements is successful.
In our look at some sports we are going to use some simple mathematical modelling to capture the essentials of events like sprinting, high jumping, long jumping, and throwing in order understand the key features and evaluate what influences, like wind assistance in the sprints, we need to worry about. In these events we will be looking at the motion of projectiles under gravity and the effects
of resistance. We will also explore some very small effects to see whether timekeepers should worry about them.
Next, we are going to look at how strength increases with size and weight. This is a general principle which allows us to understand all sorts of things in the world around us - why trees and mountains are not vastly taller than they are - as well as helping us to understand why the world records in strength events like weight lifting follow the patterns that they do as we go from one weight
category to the next. In our search for interesting small effects we are also going to see how the variation of the acceleration due to gravity (g) around the Earth's surface has an effect on events involving jumping, throwing, and lifting.
The last collection of problems we are going to look at involve probability. We will see how the sequences of actual results in the English football league compares with those that would be found if the games followed simple rules for random outcomes. You can try the same ideas out on leagues for other sports in your own country. As a follow up, you can explore some of the interesting
probability problems that arise in games like squash and volley ball where points are only scored when they are won by the serving player.
I hope that as a result of talking part in this videoconference you will come to appreciate how simple mathematics can help you understand more about what is going on when you watch sports events and even help you improve your own performance in sports as well as in mathematics. We have only got time to look at a few sports but there are interesting mathematical aspects of many other sports and I
have listed a few books which will enable you to begin exploring them a little further.
Some Further Reading
(Out of print books are available from libraries)
John Wesson, The Science of Soccer, IOP Publishing, Bristol, 2002
John Haigh, Taking Chances, Oxford Univ. Press, Oxford, 1999
Neville de Mestre, The Mathematics of Projectiles in Sport, Cambridge Univ. Press, Cambridge, 1990.
Vincent Mallette, The Science of the Summer Games, Charles River Media, Mass. USA, 1996 (out of print).
C.B. Daish, The Physics of Ball Games, English Univ. Press, London, 1972 (out of print).
Robert Adair, The Physics of Baseball, Harper, New York, 1994.
Theodore Jorgensen, The Physics of Golf, Springer, New York, 1994.
Robert Banks, Towing Icebergs, Falling Dominoes and Other Adventures in Applied Mathematics, Princeton Univ. Press, New Jersey, 1998.
Thomas McMahon, James Bonner, & John T. Bonner, On Size and Life, Scientific American library, WH Freeman, San Francisco, 1985 (out of print).