# 1.8.8 Reading distance-time graphs: summing up

You should now be able to interpret distance-time graphs, and be able to use them to find information about the average speed, the distance travelled and the time taken for different sections of a journey. Given any two of these quantities you should be able to identify and use the appropriate formula to find the third.

An important feature of a straight-line graph is its gradient. The gradient, or slope, of a graph expresses a relationship between a change measured along the horizontal axis and the corresponding change measured along the vertical axis. The steepness of the slope indicates how fast the variable represented on the vertical axis is changing with respect to the variable on the horizontal axis. So a steep slope represents a rapid *rate of change* of one variable with respect to the other, and a more gentle slope represents a lower rate of change. On a distance-time graph, the gradient is the rate at which distance is changing with respect to time. In other words, the rate of change of distance with time is a measure of speed.

## Activity 19: Impossible journey?

Figure 48 shows three distance-time graphs. For each graph, explain using brief notes, whether or not it represents a possible journey.

### Discussion

Figure 48a suggests a period of no movement, followed by an instantaneous change in distance, followed again by no movement. This distance-time profile (the general shape is sometimes called a ‘step change’) is impossible. All movements take some time to complete, you cannot travel a distance in no time at all.

In Figure 48b, the journey starts at a steady speed. This is followed by a brief period during which the speed is zero, and hence the slope of the line is zero. The next section suggests that a further distance is travelled, but time appears to be reversed, so that the traveller arrives back at the same time they started, but at a different place. Fine for time travellers, but impossible in practice.

Figure 48c shows a perfectly reasonable distance–time graph. The journey starts at a constant speed and distance increases with time. The direction of travel then changes, and the traveller begins to return towards the starting point at a steady speed. Direction changes again and the graph indicates a steady speed away from the starting point during the final section.