Exploring distance time graphs

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# 1.8.6 The final graph

The three separate lines are combined into one overall distance-time graph representing the entire journey, as shown in Figure 44. The times for the sections are added together, so that the scale on the horizontal axis shows the total time that has elapsed since leaving Paris. Similarly, the distances of the sections are combined, so that the scale on the vertical axis shows the total distance from Paris. On the graph, the point (46, 230) represents the ending of the journey across northern France and the start of the journey through the tunnel, and the point (66, 280) represents the emergence of the train from the tunnel and the start of the final part of the journey into London.

Figure 44 The overall Paris-London distance-time graph

The first thing to notice about the overall distance-time graph is that it is not one straight line, so it cannot be represented by a simple proportional relationship, although it was built up by looking at the proportional distance-time relationships for each section. There are three straight line sections to the graph. Look at the slopes. The slope or gradient of the first section, representing the journey from Paris to the tunnel, is the steepest of the three, indicating that the train travels at its highest speed across northern France. The train slows down for its 50 km journey through the tunnel, its lower average speed represented by the shallower slope of the central straight-line section of the graph. Emerging from the tunnel, the train slows down further for its journey across southeast England into London. This final section of the graph has the smallest gradient (slowest speed) of all. In each of the three sections of Figure 44, the slope of the graph differs from the average slope. This means that the average speed of the train for each of these three parts differs from its overall average speed. From Paris to the tunnel the speed is considerably greater than the average. Through the tunnel the speed is slightly more than the average, while through south-east England the speed is less than the average.

## Activity 16: Interpreting a distance-time graph

Rana delivers newspapers. Figure 45 is a distance-time graph of her round. Using the graph, answer the following questions.

1. On which section of the journey do you think she was walking the fastest? Make some brief notes to explain your answer.

2. What is your interpretation of section CD?

3. Part of her round is up a hill where her walking speed is slowest. Which section of the graph do you think represents this?

4. Which part of the graph represents Rana’s return to her starting place. Why?

### Discussion

To tackle this activity, you need to be able to estimate Rana's walking speed as she completes her round. Her speed at any point is indicated by the slope of the distance-time graph. A steep slope means she was walking quickly and covered the distance in a relatively short time. A gentle slope means she was walking relatively slowly and took longer to cover the distance. A slope of zero—where the graph is horizontal—represents zero speed, indicating that she had stopped.

Rana was walking fastest on section BC. Here, the slope of the graph is steeper than in any other section, indicating the highest speed.

Section CD is horizontal indicating Rana's speed was zero for a short time. She takes the break at the furthest distance from the start.

Assuming Rana walks slowest when she is going up the hill, this section is represented by AB.

Section DE represents Rana's return. The gradient is negative because the distance from the start is decreasing with time, as Rana completes the round and returns back along the same route to her starting point.

Figure 45 Distance-time graph of Rana’s round
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