Describing motion along a line

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# 2 Positions along a line

## 2.1 Simplification and modelling

Everyday experience teaches us that unconfined objects are free to move in three independent directions. I can move my hand up or down, left or right, backwards or forwards. By combining movements in these three directions I can, at least in principle, move my hand to any point in space. The fact that there are just three independent directions, and that these suffice to reach any point, shows that the space in which my hand moves is three-dimensional.

The motion of a large object, such as an aeroplane, moving in three-dimensional space is very difficult to describe exactly. The aeroplane may flex, rotate and vibrate as it moves, and there may be complicated changes taking place within it. To avoid such complexities at the start of our investigation of motion we shall initially restrict our attention to objects that move in just one dimension along a line.

We shall treat the object concerned as a particle, that is, a point-like concentration of matter that has no size, no shape and no internal structure.

Treating a real object, such as an aeroplane, as though it is a particle is clearly a simplification. Real objects certainly do have size, shape and internal structure, but such details can often be neglected in specific contexts. Making simplifications of this kind is an important part of the skill of scientific modelling in physics. A good model uses the well-defined concepts of physics to represent the essential features of a problem while omitting the irrelevant details. The trick is not to oversimplify. The model should be as simple as it can be, but no simpler. Just what this entails will depend on the problem being analysed. For example, the use of the year as a unit of time is a result of the orbital motion of the Earth around the Sun. This orbital motion is described quite easily while treating the Earth as a particle. The Earth's diameter is about 10 000 times smaller than the distance between the Earth and the Sun, so a particle model is a very good approximation in this case. However, a particle model of the Earth cannot account for the distinction between day and night since that depends on the rotation of the Earth.

In this course we shall only consider problems that can be adequately modelled by particles moving in one dimension, that is, along a straight line.

Describing the motion of a particle moving along a line may sound like a fairly simple undertaking, but, as you will see, it will present plenty of challenges and will allow us to gain significant insights into the operation of systems such as zero gravity drop-towers and vertical-drop roller-coasters.

### Question 1

List some more examples of real motions that might, in your opinion, be reasonably well modelled by particles moving along a line.

Your list might well include items such as: the motion of a passenger on a train, or in a plane or in any other vehicle, as long as it is the passenger's overall position that is important, and not their posture or internal movement. You might also have listed the vehicles themselves, provided the same conditions apply. Indeed, you might list almost anything, including the Earth or the Sun, provided you are considering a context in which the moving object can be treated as point-like.

The branch of physics that is concerned with the description of motion is known as kinematics. Kinematics is not concerned with forces, nor with the causes of motion; those topics are central to the study of dynamics. Typical questions that we might ask about the kinematics of a particle are:

• Where is the particle?

• How fast is it moving, and in what direction?

• How rapidly is it speeding up or slowing down?

How such questions are to be answered is the main concern of the rest of this course.

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