Exploring distance time graphs

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

# 1.4 Graphical conversions

## 1.4.1 Introduction

The term ‘conversion graph’ describes a graph used to convert a quantity measured in one system of units to the same quantity measured in another. For example, you can draw up a conversion graph to convert temperatures expressed in degrees Celsius to temperatures expressed in degrees Fahrenheit; to convert liquid volumes expressed in pints to the same volumes expressed in litres; to convert a sum of money expressed in one currency to the same amount expressed in a different currency.

One way to convert one measurement into another is by using mathematics. In this case you need to use the map scale to convert between distances and areas on the map and on the ground. The relationship you need has the general form

measurement on the ground = some number x measurement on the map.

For your 1 : 25 000 map, the relationships between distances and areas are

distance on the ground = 25 000 x distance on the map

and

area on the ground = (25 000)2 x area on the map

where, in each case, the map and ground measurements used the same units.

Now these relationships are expressed as formulas, but you can represent the same information graphically. Representing things in a different way can offer a new perspective and a new way of thinking about a relationship. Often it is helpful to move between different viewpoints and different representations when you are trying to understand a problem. Figure 7 shows the relationship between distances in the form of a graph.

Figure 7 Graph of ground distance against map distance

Look at the way Figure 7 has been constructed. By convention, the map distance – which is what you know – is plotted along the horizontal axis, and the ground distance – which is what you want to find out – is plotted along the vertical axis. The axis of a graph is rather like a map. The axis represents a quantity (such as temperature, distance, time). The scale of the axis, like the scale of a map, relates the distance along the axis from the origin to the amount of the quantity. Notice the scales and the labels on the axes. Map distances are conveniently measured in centimetres, but ground distances are more conveniently quoted in kilometres. So in this case the scale on the horizontal axis relates to a measurement in centimetres and the scale on the vertical axis relates to a measurement in kilometres.

The labels on each axis tell you about the thing that is measured, and the units it is measured in. So the horizontal axis is labelled ‘Map distance/cm’, and the vertical axis is labelled ‘Ground distance/km’. Notice that there is an oblique line between the quantity and its units: ground distance divided by kilometres, and map distance divided by centimetres. This means that the numbers along the horizontal and vertical axis are ratios – pure numbers – which can be added, subtracted, multiplied or divided without worrying about their units.

Remember that quantities with units are the result of measurements taken in the real world. When you draw a graph you are not dealing with actual kilometres or centimetres but with their representations as lengths along the axes of the graph. You can perform calculations with the numbers to make predictions about the material world. But when you go back to that world, remember to restore the units, so that the numbers relate once again to actual measurable distances.

Returning to the formula for the conversion graph, recall that the starting point is a scale relationship which links map distance to ground distance, when both are measured in the same units. If the units are centimetres, you have

ground distance in centimetres = 25 000 x map distance in centimetres

But how to convert from a map distance measured in centimetres to a ground distance measured in kilometres? This changes the formula slightly.

There are 100,000 centimetres in a kilometre, so to convert from centimetres to kilometres divide by 100,000. The formula becomes:

Whatever the map distance is in centimetres, the ground distance in kilometres can be found by multiplying by 0.25. Mathematically, the ground distance is said to be directly proportional to the map distance. This means that ground distance is found simply by multiplying the corresponding map distance by some fixed number, known as the constant of proportionality. Another way of expressing this is the fact that the ratio of ground distance to map distance is constant (and equal to this constant of proportionality).

ground distance (in km) / map distance (in cm) = 0.25

MU120_3