1.8.9 A mathematician’s journey
Distance-time graphs can show more than one journey on the same graph. The journeys do not have to start from the same place, or start at the same time, but all times and distances must be measured from a common origin along a common route. In this subsection, you will see how drawing a distance-time graph can help in planning a journey.
Bob and Alice both work for the Open University. Bob lives in Edinburgh in Scotland and Alice lives in Milton Keynes in England about 510 kilometres to the south. During a phone call they discover that Alice will be travelling 340 km north to the Newcastle Regional Centre and Bob will be travelling 420 km south to an OU summer school in Nottingham on the same day. They both aim to start their journeys at about 10 am.
Figure 49 shows a network map of their journeys, indicating the distances between the cities. A network map ignores all features of a journey except place names and distance.
It is unlikely that Alice would actually drive through Nottingham on her way to Newcastle. However, Nottingham lies very close to the motorway she would be using. For this model, ignore this detail, and assume Nottingham is directly on Alice’s route.
Will they pass each other on the road going in opposite directions, and if so, can they arrange to stop and meet at a convenient point? Alice – a mathematician – offers to draw up a distance-time graph to model the journey. Mathematical models can be used to describe events that have occurred, or to predict how events will go in the future. Alice will use her model to predict how the actual journeys might go. To get started, she needs to make some assumptions. She can build these into her model and then check to see whether the predictions it gives seem reasonable.