Making simplifying assumptions

When setting up a model, you first need to decide what is essential. For example, what would you take into account and what would you ignore when considering the motion of a car travelling from San Francisco to Los Angeles?

You will need to know the distance and the time taken for parts of the journey, but you might decide to ignore the dimensions of the car and the motion of the wheels. You would then be using the idea of a particle to model the car. A particle has no dimensions.

You might also decide to ignore the bends in the road and its width, and so treat it as a straight line with only one dimension. A length along the line would represent a length along the road in the same way as a piece of thread following a road on a map might be straightened out to measure its length.

You might decide to split the journey up into parts and assume that the speed is constant over these parts. The process of making decisions like these is called making simplifying assumptions and is the first stage of setting up a mathematical model of the situation.

Defining the variables and setting up the equations

The next step in setting up a mathematical model is to define the variables with suitable units. These will depend on the problem you are trying to solve. Suppose you want to know where you ought to be at certain times in order to maintain a good average speed between San Francisco and Los Angeles.

You might define your variables as follows:

• the total time since the car left San Francisco is t hours
• the distance from San Francisco at time t is x km
• the average speed up to time t is v km h –1.

Then, at Kettleman City t = t1 and x = x1; etc.

You can then set up equations and go through the mathematics required to solve the problem. Remember to check that your answer is sensible. If it isn’t, you might have made a mistake in your arithmetic, or your simplifying assumptions might need reconsideration.

The theories of mechanics that you will learn about in this course, and indeed any other studies in which mathematics is applied, are based on mathematical models of the real world. When necessary, these models can become more complex as your knowledge increases.

For a much shorter journey, you might need to take into account changes in the speed of the car. This chapter develops the mathematics required when an object can be modelled as a particle moving in a straight line with constant acceleration. In most real situations this is only the case for part of the motion – you wouldn’t expect a car to continue accelerating at the same rate for very long – but it is a very useful model to use as a first approximation over a short time.

(Ref: Cambridge International AS and A Level Mathematics by Sophie Goldie, Series Editor: Roger Porkess, Hodder Education)