Exploring distance time graphs

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1.8.3 Distance, speed and time: assumptions

The formulas for speed, distance and time are all examples of mathematical models. Here, you should bear in mind that such models stress some aspects of travelling but ignore others. Building a mathematical model involves making some assumptions, and usually this involves disregarding those inconvenient aspects of real-world events which can not easily be fitted into a mathematical description.

Take, for example, the model s = d/t used to calculate speed. Dividing a journey distance by the travelling time gives a single number which represents the average speed on the journey. The formula contains no information about the style of transport, about the joys, delights, delays and frustrations of travelling, about stops for petrol or children being sick. The typical complexities of even an everyday journey have been boiled down to just two numbers – the overall distance and the total time taken.

The relationship between distance, speed and time can serve as the basis for representing a journey as a graph. Recall that it is:

distance = average speed x time

You should recognise this formula as a directly proportional relationship. The constant of proportionality in the relationship is equal to the average speed. At any particular average speed, the distance travelled is directly proportional to the time the journey takes.

• If you plot a graph of distance travelled against time, what sort of graph will you get?

Look at Figure 37. The vertical axis represents distance travelled along the route and the horizontal axis represents time. Both are measured from the start of the journey. For any particular average speed, the graph of distance against time is a straight line starting at the origin. The average speed is represented by the gradient, or slope, of the graph. This type of graph is called adistance-time graph.

A graph like this is described mathematically by the following ‘straight-line’ relationship.

value on vertical axis = gradient x value on horizontal axis

Figure 37 Distance-time graph
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