Overcoming Yaw

When a non-spherical projectile is fired or launched its longitudinal axis rarely lies along the tangent to its trajectory, so there is a non-zero angle of yaw. As a result of yaw the air acts on the projectile to produce

1. (i) an increased drag compared with the drag at zero yaw,

2. (ii) a deviating force which tends to alter the projectile's trajectory,

3. (iii) an overturning moment which rotates the projectile about its centre of gravity.

The last was discussed in Section 6.7, and may increase or decrease the yaw depending on the design of the projectile. A decrease in yaw is obtained by fitting fins to some projectiles (darts, arrows, bombs and mortar shells), so providing a method of stabilising the projectile and its trajectory. This method of stabilisation has some disadvantages however, the principal one being that a cross-wind tends to interact with the fins and push the projectile well off course.

Effects Other Than Constant Gravity and Variable Drag

The basic equations (4.1) or (4.3) for a projectile are essentially restricted to two dimensions. If an OZ axis is introduced orthogonal to the OX and OY axes then the absence of forces in the z-direction combined with the initial conditions z = dz/dt = 0 when t = 0 ensures that z ≡ 0 for all t. However, any non-symmetrical variation (such as spin or yaw) in the z-direction will produce a drift effect on the projectile in this lateral direction.

As soon as spin is imparted to a projectile to reduce yaw and impose stability of flight there are many effects such as cross-forces and cross-torques which have to be included in the equations of motion (see Chapter 7). Even in the absence of spin the non-symmetrical shape of some projectiles or the yawing of a symmetrical projectile will give rise to a cross-force known as lift.

The spinning of the earth also requires the inclusion of pseudo-forces (Coriolis and centrifugal) in the governing equation, particularly when the projectile has a very long flight path. For these problems gravity variations due to altitude and the non-spherical shape of the earth may also be important.

Classes of Projectiles

So far a number of mathematical techniques have been considered that can be used to calculate projectile trajectories. With this background the particular techniques will be identified that can be applied to various projectile problems that arise in sports and recreational activities. The list of projectiles considered is not exhaustive, but is wide-ranging enough to show the general approach required if the reader wants to calculate the trajectory for a projectile that is not specifically considered in this chapter.

Two classes of projectiles will be considered; those in which the human body is the projectile and can influence the trajectory during the time of flight, and those in which the human body propels the projectile and has no further influence on the trajectory. For athletic field events the former would include the long jump, triple jump, high jump and pole vault, while the latter covers the javelin, discus, shot-put and hammer throw. The throwing events are therefore less complicated during the projectile phase, but probably more complicated during the build-up to the release of the projectile, and have been surveyed in detail recently by Hubbard (1988).

Many non-human projectiles are spherical, but other shapes will be considered also, particularly those that are streamlined.

A large number of sporting events contain the motion of a projectile; yet research papers have appeared only intermittently on the mathematical techniques associated with projectiles in sport. A recent book by Hart and Croft (1988) presents a subset of these techniques for a small selection of events. The aim of the current book is to present a unified collection of the many problems that can be tackled and of all the mathematical techniques that can be employed.

The mathematical foundations of the subject of projectiles were developed from investigations of the motion of bullets and shells. There are some excellent texts describing this research in detail, the best being by McShane, Kelley and Reno (1953). Emphasis here will be on the non-military applications of the behaviour of projectiles in flight, which have received only limited attention.

The purpose of the book is to collect together the various mathematical tools and techniques that will help the reader solve many projectile problems associated with sport or recreation. It begins at an undergraduate level, and emphasises the usefulness of this special topic as a way of teaching mathematical modelling (de Mestre, 1977). A basic knowledge of classical dynamics, calculus, vectors, differential equations and their numerical solution is assumed. At the end of each chapter are exercises, many of which lead on to suitable research projects for honours or masters students.

The Basic Equations

The air resistance acting on a projectile can be separated into four categories – forebody drag, base drag, skin friction and protuberance drag. Forebody drag arises because some of the energy of the projectile is used to compress the air in front of the projectile and in some cases to form shock waves. Base drag is due to the turbulent wake behind the projectile and is more pronounced if the rear of the projectile is blunt. Skin friction is caused by air adhering to the surface of the projectile. It can be reduced by polishing the surface. The protuberance drag is really a combination of the forebody drag, base drag and skin friction on any protuberance attached to the main projectile body. For example the principal protuberance on a shell fired from a gun is the driving band.

When the projectile's speed (v) is divided by the local speed of sound in air (α) this nondimensional ratio is called the Mach number (M). Near M = 1 (the transonic region) the forebody drag increases dramatically, and therefore becomes effectively the total aerodynamic drag. Since a is proportional to the square root of the absolute temperature of the air, and the drag coefficient CD introduced at the beginning of Chapter 3 also depends on air temperature, it follows that CD is a function of the Mach number for a given projectile (see Figure 4.1).

Velocity and Position Vectors

The formulae derived in Chapter 1 relate strictly to a projectile travelling under the influence of constant gravity in a vacuum. When the projectile moves through any fluid medium (gas or liquid) other forces are present due to the slowing-down influence of that medium's particles. The sum of these forces in a direction opposite to the projectile's velocity is called the drag force and for many non-spinning projectiles it is the main effect of the medium. A more detailed discussion of drag will be postponed until the beginning of Chapter 4.

Experiments show that this drag force is usually related in a non-linear way to the velocity of the projectile. When a projectile is moving at moderate or high speeds the non-linear drag force can be approximated by using different powers of the speed over different velocity ranges.

For projectiles moving through air at very low speeds, or for motion through other fluids where the Reynolds number (see Chapter 7) is small, the drag can often be assumed to be directly proportional to the speed. This linear model shows the effect of the inclusion of drag without the mathematics becoming too complicated at first.

The last two chapters have explored something of the background to, and the practicalities of, mathematical modelling. This book, however, is concerned primarily with one particular technique which has been used to develop, in students, the skills of mathematical modelling. In this chapter the development of the case study/simulation method is traced and its use by the author in his own teaching activities is described.

The context of the case study course

At the time when the course was designed, created and first used the Faculty of Engineering at Bristol University comprised five departments, Aeronautical Engineering, Civil Engineering, Mechanical Engineering, Electrical and Electronic Engineering and Engineering Mathematics. Historically the role of the Engineering Mathematics Department had been to meet the mathematics teaching requirements of the four engineering departments. Such an arrangement has considerable advantages over the possible alternatives whereby the mathematics teaching needs of engineering students are met either by service mathematics courses given by the Mathematics Department of the institution concerned or by the engineering departments themselves. In the first case there is often a strong feeling that mathematics lecturers have insufficient appreciation of and sympathy for the special needs of engineers resulting in rather theoretical mathematics courses which do not engage the interest of engineering students who, after all, are primarily interested in the applicability of the mathematics to their own disciplines. In the second case there is a tendency for the engineering lecturers to teach mathematics as a series of recipes for solving particular engineering problems with insufficient attention to the overall structure of mathematics as a subject in its own right.

In chapter one it was suggested that a degree course in mathematics should seek to teach firstly a body of mathematical knowledge, secondly the ability to extend that knowledge independently and thirdly the ability to use that knowledge. It was also suggested that mathematics courses in tertiary education have traditionally concentrated on the first of these to the detriment of the other two. There is however, within the discipline of mathematical education today, an identifiable group of teachers and lecturers who, to a greater or lesser extent, believe that the teaching of mathematical modelling is vital to the development of an ability to use mathematics and that such teaching is necessarily a distinct activity from the teaching of other topics in mathematics. The existence of such a group, who might be characterised as the mathematical modelling movement, is a distinctively recent innovation in mathematics – it would have been difficult, for instance, to identify any such grouping prior to the mid nineteen sixties. We might ask what circumstances have led to the birth of the movement and why has its development been so rapid? In this chapter the origins of the movement will be described, the development of the theory of mathematical modelling traced and some new insights into the nature of the mathematical modelling process proposed.

The need to teach mathematical modelling

A panel discussion, organised in 1961 by the [American] Society for Industrial and Applied Mathematics, between Professors Carrier, Courant, Rosenbloom and Yang entitled ‘Applied mathematics: what is needed in research and education’ is reported by Greenberg (1962).

This book is the culmination of more than ten years work by the author on ways of teaching undergraduate students how to become not only competent mathematicians but also skilled users of mathematics in the solution of problems arising in the real world.

In a degree course, at least in mathematics and those related disciplines in which mathematics is a tool, three things must be taught. Firstly, there is a body of factual knowledge and technique which students must acquire. Secondly, the skill of extending their mastery of factual knowledge and technique must also be acquired, that is they must learn how to learn more mathematics as and when the need arises. Thirdly, since all the mathematical knowledge and technique in the world is of little use to the practical mathematician, engineer or scientist if the skill of applying that knowledge to their professional problems is missing, students must learn how to use their mathematical knowledge in solving the problems of the real world. In the past the second and third of these aspects of mathematics have not been formally taught, rather it was assumed that students would acquire them incidentally (one might almost say accidentally) whilst studying the facts, theorems and techniques of mathematics. Indeed there has been considerable doubt as to whether these skills could be taught effectively at all. At the same time, though, one of the classical justifications advanced for teaching mechanics has always been that, through the study of mechanics, students will learn something about the art of applying mathematical theory to the problems of the real world.

This chapter attempts to identify and illustrate some of the practical skills needed by an effective mathematical modeller. It does this partly by direct comment on the modelling process and partly through the medium of a number of examples of the development of simple mathematical models. It should be apparent that no description of the processes of mathematical modelling can ever hope to be absolutely complete or to encapsulate the ‘last word’ on the subject. In that spirit then, this chapter draws on experience accumulated over a number of years by the author both in his capacity as a mathematician working in an engineering environment using his mathematical knowledge in the solution of engineering problems and in his capacity as a teacher of mathematics and mathematical modelling.

Understanding the problem

The first general comment to be made is that very little progress can be made in modelling if the modeller does not fully understand the system or situation which he or she is trying to model. (Before going any further let us, for the sake of brevity and clarity, acknowledge that the subject of a mathematical study may be any of a wide spectrum of entities ranging from concrete physical mechanisms through to relatively abstract systems of human activities and therefore agree to denote all of these possibilities by the single term ‘subject’.) The first stage of modelling should be the collection of data or experience about the subject to be modelled.