Mathematics and Science for Exercise and Sport

The Basics

Craig Williams, David James, Cassie Wilson

• Publisher: Routledge
• Year: 2008
• ISBN:
  9781134108114
• Edition: 1
• Topic: Éducation
• Subtopic: Éducation générale

Mathematics and Science for Sport and Exercise introduces students to the basic mathematical and scientific principles underpinning sport and exercise science. It is an invaluable course companion for students who have little prior experience of maths or science, and an ideal revision aid for higher level undergraduate students.

The book explains the basic scientific principles that help us to understand sport, exercise and human movement, using a wide range of well-illustrated practical examples. Written by three leading sport scientists with many years experience teaching introductory courses, the book guides beginning students through those difficult to grasp areas of basic maths and science, and identifies the common problems and misconceptions that students often experience. It includes coverage of key areas such as:

• science of physical states – gas, liquid and solid
• science of biomechanics, motion and energy
• mathematical formulae, calculus, and differential equations
• statistics
• scientific report writing
• key concepts such as pressure, torque and velocity
• self-test features and highlighted key points throughout each chapter.

Fully referenced, with guides to further reading, this book is an essential companion for all students on foundation or undergraduate level courses in sport and exercise science, kinesiology, and the human movement sciences.

PART 1
PHYSICAL STATES

CHAPTER ONE
INTRODUCTION

CHAPTER CONTENTS

Aims of the chapter

Introduction

Science and maths in exercise and sport

Measurement

Making measurements on humans

Conclusion

Key points

Bibliography

Further reading

AIMS OF THE CHAPTER

This chapter aims to introduce science as applied to exercise and sport. After reading this chapter you should be able to:

• Understand the basis for maths and science in exercise and sport.
• Recognise the Système Internationale for units of measurement.
• Understand the importance of accuracy of measurement.
• Recognise scientific notation.
• Recognise issues relating to measurements on human participants.

INTRODUCTION

The study of exercise and sport draws upon many disciplines, including biomechanics, medicine, nutrition, philosophy, physiology, psychology and sociology, to name a few. In the sport performance area, biomechanists, physiologists, psychologists, medics and nutritionists often work alongside athletes and coaches with the aim of improving performance. In the exercise and health area, these scientists work alongside patients/ healthy clients and exercise and health practitioners with the aim of improving health. Whatever the aim of the interdisciplinary team, the quality of the science underpinning practice is critical to reaching the ultimate goals. In turn, many scientific approaches require a sound appreciation of basic mathematics.

Much of the work of members of an exercise or sport team should have a sound scientific basis. It is a key responsibility of the exercise or sport scientist to ensure that the approach has a scientific basis. In practice, this might be evident through a systematic approach to solving problems, or ensuring that high-quality evidence exists for implemented strategies.

It has been said that, ‘there is no such thing as applied science only the application of science’ (Huxley). This quotation applies to the area of exercise and sport science in that, to investigate an exercise or sport problem, scientific principles are often applied to the problem. In this respect, appreciating what a scientific approach may bring to a situation is of benefit to all members of an exercise or sport team.

SCIENCE AND MATHS IN EXERCISE AND SPORT

A scientific approach to the study of exercise and sport is often considered useful, particularly when the benefits of exercise or sport participation must be maximised. The scientific body of evidence that should underpin practice is based on a set of principles that are widely understood by the scientific community. A scientific approach involves posing a problem, often in the form of a question, which can then be investigated. Once the question has been investigated, the results may be used to develop a theory. The theory may then be applied to similar problems in the future. However, theories evolve as they are challenged. If it is found that the theory does not apply to a certain situation, the theory must be modified to account for that situation. It is the responsibility of scientists to continually challenge theories so that they can be either modified or discarded. It is these principles that guide work in science. Williams and Wragg (2004) provide a brief overview of the development of scientific theory, and consider alternative paradigms.

Existing knowledge in exercise and sport science is often challenged through the collection of new factual information. Prior to the collection of such information, it is necessary to formulate a precise question. The question is normally posed in the form of hypotheses that can be tested through the collection of information. Two hypotheses are normally formulated, based on two possible outcomes. One hypothesis is known as the null hypothesis, and relates to a theory remaining unchanged. The other hypothesis is known as the alternative hypothesis, and relates to a need for modification of a theory. At the end of the investigation, one hypothesis is rejected and one is accepted.

The information that is collected in order to test the hypotheses is normally referred to as data. In Chapter 8 the close relationship between data collection, data analysis and hypothesis testing is examined in more detail.

MEASUREMENT

The collection of data often involves the measurement of some related phenomenon. Measurement might appear an easy task, but on closer inspection it is considered to be an involved process. The quality of measurement is critical in science, so the remainder of this chapter will examine aspects of measurement. Measurement concepts will underpin many of the following chapters, hence their inclusion at this early stage of the book.

A student of exercise and sport should have a good understanding of measurement issues. A common problem related to the understanding of measurement is described by Paulos (1988). In an attempt to familiarise students with numbers and what their quantity means, Paulos asked a student how fast (in miles per hour) human hair grows, to which the student replied, human hair doesn’t grow in miles per hour! The answer is actually 10^–8 miles per hour or 0.00000001 miles per hour. In this example, the student failed to realise that hair growth, albeit very slow, could be expressed as miles per hour as a unit of speed.

To understand measurement units, it is necessary to learn about Système Internationale (SI) units. The SI units are an abbreviation of ‘le Système Internationale d’Unités’. These standardised units have been developed to promote international co-operation and to provide a universally accepted system of measurement. The system not only allows information from different countries to be easily exchanged and understood, but also offers a standardised form for presentation of measurement outcomes. The seven base units of the SI are found in Table 1.1. It is important that the fundamental units are learnt because other units are subsequently derived from these seven.

Table 1.1 The seven base SI units

The symbols are a mixture of lower-case and upper-case letters, which can be confusing. All symbols are written in lower-case roman letters except when the name of a unit is derived from the name of a person; for example, (W) to symbolise power in watts. When the unit has a proper name and is written in full, the whole word is in lower case. For example, the unit of measure for pressure is ‘Pa’, named after the scientist Blaise Pascal (1623–1662); when written out in full, it should be written as ‘pascal’. A common mistake made by students is the symbol representing kilogramme (for measurement of mass), often writing it as ‘Kg’; this is incorrect as it should appear as ‘kg’. Recall, from Table 1.1, that the upper-case ‘K’ is the symbol for temperature, named after Lord Kelvin (1824–1907). All symbols should be lower case, except where the unit is derived from a proper name. An exception to the rule is the litre, where it is now common practice to accept it as ‘L’. This exception is partly to avoid confusion between the lower-case ‘l’ and the numeral ‘1’.

Other rules include never following a period after a symbol unless at the end of a sentence, as well as never pluralising any symbols. Another common mistake is to mix names and symbols together; for example, newton·metre·s^–1. The correct style should be N·m·s^–1. Students are also sometimes confused as to when to use the solidus symbol (/). With personal computers it should be possible to learn how to use the symbol period instead, that is, · raised above the line when there is a product of two units. The solidus can be used but is not considered to be as clear for scientific work, especially when multiple symbols are combined. If the solidus symbol is used then only one per expression should be included, that is, kg·m/s², this is because the division is not associative. When two or more units are formed by multiplication or division in text, a multiplication of two units is indicated by a space between two words and never by a hyphen; for example, newtonmetre would be incorrect; rather, newton metre is the correct style. To indicate in text a division of several units we would write ‘per’, rather than use the solidus; for example, litres per minute rather than litres/minute. When units are reported, it is preferable to use symbols rather than writing out the full name. Whenever numbers are reported with symbols, a space between the two should be left, that is, 400 W rather than 400W. There is, however, an exception to the rule, that is, the use of the symbol for degree (°). In this instance, there is no space left between the numerical value and the symbol: 40° not 40° and 30°C not 30 °C.

When reporting measurement quantities, a zero should always be placed before a decimal, that is, 0.01 not .01. Decimals are preferable to fractions, that is, 0.75 is preferable to ¾. If possible, long numbers should be separated with the use of a space, so 1,300,000 becomes 1 300 000. It is, however, optional with four digits; for example, 1000 or 1 000.

Derivatives of units in Table 1.1 are included in Table 1.2. Students should be aware that there are a number of other non-SI units associated with time. Although the SI unit for time is the second (s), day (d) is sometimes accepted for long periods of time. In Table 1.3 are the non-SI units for time, commonly used in science journals.

Other derivatives from the original seven base units (Table 1.1) such as energy, power and force will be dealt with in greater detail in Chapters 5 and 6.

Table 1.2 Units derived from the seven base SI units

Table 1.3 Symbols for the non-SI units for time

One of the reasons why the SI was devised was to help to report data that was numerically very large or very small. Therefore, a number of prefixes are used to form multiples and sub-multiples of the base units. These prefixes usually change the quantity by a factor of 10³ or 10^–3, but other smaller or larger increments are used (see Table 1.4). The prefix is written without a space and the symbols are as shown in Table 1.4. Therefore, a megagram (Mg) is a kilogramme (kg) multiplied by a thousand.

It is often impractical to write a number with several zeros if you are dealing with very large or small numbers. To overcome this issue, scientific notation, or powers of ten, can be used. It is common practice to shift the decimal place between the number one and ten and then multiply by the appropriate number. For example, if you wanted to write the number of thin filaments in a single muscle fibre, which may be about 64 000 000 000, on several occasions, it would become very tedious to constantly write this expression. By using notation, you can convert this number by placing a decimal place between ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• List of figures
• List of tables
• Preface
• Acknowledgements
• Part 1 Physical states
• Part 2 Force, pressure, energy and electricity
• Part 3 Scientific transferable skills
• Appendix 1 Health questionnaire
• Appendix 2 Example consent form
• Appendix 3 Thermal equivalents of oxygen
• Appendix 4 Scientific journals in exercise and sport
• Appendix 5 Measurement concepts
• Key terms