1.8.11 A mathematician’s journey: using the model for planning
By drawing a distancetime graph, Alice has predicted that she and Bob will pass on the stretch of road between Newcastle and Nottingham. Using the OU’s computer system, she sends an email message to Bob suggesting that they meet at a roadside restaurant about 275 km north of Milton Keynes (for Bob this will be 510 − 275 = 235km south of Edinburgh). Bob acknowledges her email and the meeting is set up.
Alice guesses they will probably stop for about 30 minutes. But what effect will this have on the times they will reach their respective destinations? She can modify her graphical model to include the stop to predict the consequences.
Figure 51 shows the modified distancetime graphs. Bob’s graph shows that he will probably arrive at the restaurant first at about 2.00 pm. Alice is likely to arrive about 20 minutes later. If they stay for 30 minutes and both leave around 2.50 pm, Bob will have been there for nearly 50 minutes. So he is likely to complete his journey about 50 minutes later than he originally planned. Since Alice made only a 30minute stop, she should get to Newcastle at about 4.00 pm, 30 minutes later than she had originally planned.
Activity 21: Using the model for planning

How would you use Figure 51 to find out what Alice’s average speed after her break should be if she wants to arrive at the restaurant at the same time as Bob? Note down the steps you would take to find the answer.

What will be the average speeds for Bob and Alice’s complete journeys if they keep to their plan?
Discussion

To arrive at the same time as Bob, Alice must cover the 105 km in 90 minutes, or 1.5 hours. So her average speed must be 105/1.5=70 km per hour.

Alice covers her journey of 340 km in 6 hours, so her average speed (overall) is 340/6=56.7 km per hour. Bob completes his journey of 420 km in 7 hours 50 minutes, or 7.83 hours. So his average speed (overall) is 420/7.83=53.6 km per hour.