1.8.10 A mathematician’s journey: building a model
She estimates she will drive for about two hours at an average speed of about 85 kilometres per hour and then stop for a break for about 30 minutes. She then intends to continue her journey to arrive finally in Newcastle around 3.30 pm. Alice also needs Bob’s estimates. He reckons to drive for about 2.5 hours to cover 170 kilometres, stop for about 30 minutes and then press on for Nottingham to get there at about 5 pm.
Alice summarises their planned journeys in two tables. She measures times from 10 am, and distances from Milton Keynes. Table 6 shows Alice’s journey times (in hours) from her 10 am starting time, and the distance (in kilometres) she has travelled from Milton Keynes. You can see that at the start of the journey she will be 0 km from home, and two hours later she will be 2 × 85 = 170 kilometres away. The distance does not change while she has her break. After her break, the remaining 170 km to Newcastle should take about three hours at an average speed of 170/3 = 56 km per hour.
Table 6 Alice's journey
Time of day  10.00 am  12 noon  12.30 pm  3.30 pm 

Time after start (hours)  0  2  2.5  5.5 
Alice’s distance from Milton Keynes (kilometres)  0  2 × 85 = 170  170 + 0 = 170  170 + (3 × 56) = 340 
Now look at Table 7 showing Bob’s predicted journey. He also starts at 10 am and aims to cover 170 km in about 2.5 hours, an average speed of 68 km per hour. After a 30minute stop near Newcastle, he reckons to drive the remaining 250 km to reach Nottingham four hours later, at about 5 pm. To achieve this his average speed must be 250/4 = 62.5 km per hour. Bob will be driving away from Edinburgh and towards Milton Keynes (although his destination is, of course, Nottingham), so his distance from Milton Keynes will decrease with time.
Table 7 Bob’s journey
Time of day  10.00 am  12.30 pm  1.00 pm  5.00 pm 

Time after start (hours)  0  2.5  3  7 
Bob’s distance from Edinburgh (kilometres)  0  2.5 × 68 = 170  170 + 0 = 170  170 + (4 × 62.5) = 420 
Bob’s distance from Milton Keynes (kilometres)  510  510 −170 = 340  340  510 − 420 = 90 
Figure 50 shows the distancetime graphs for Alice’s and Bob’s proposed journeys.
The horizontal axis shows the time of day, and the vertical axis shows the distance in kilometres from Alice’s starting place in Milton Keynes along the route. Milton Keynes is the reference point for all measurements of distance. For Alice’s journey, the first part of the graph is a straight line with a positive slope (the line slopes up from left to right). The gradient of the line represents a steady speed of 85 km per hour. During her break, her speed will be zero and hence the graph is a horizontal line with a gradient of zero. The second part of her journey is modelled by another straight line with a positive slope, this time indicating an average speed of 56 km per hour. The positive slopes indicate that Alice’s distance from Milton Keynes will increase with time.
Bob’s distancetime graph is similar to Alice’s except that the slopes of the lines (apart from the break) are negative, they slope down from left to right. The negative slope means that Bob’s distance from Milton Keynes will decrease with time as he drives south. In other words, the gradient of a line of a distancetime graph contains information about the direction as well as the speed, of travel.
The gradient of a distancetime graph can be positive, negative or zero, depending on whether travel is away from, towards, or stationary relative to, the starting point of the journey. Speed itself, however, is always a positive quantity. What the gradient indicates is velocity. Velocity is speed in a particular direction. On a distancetime graph, a positive gradient represents a positive velocity: that is, a speed in a direction away from the place represented by the origin of the graph. A negative gradient represents a negative velocity, that is speed in a direction towards the origin.
The distancetime graph for Alice’s journey in Figure 50 shows positive gradients, indicating positive velocity, because she will be going away from Milton Keynes, the reference point for distance measurements. Bob’s graph, on the other hand, shows negative gradients, indicating negative velocity because he will be travelling towards Milton Keynes. Alice’s estimated speed during the first section of her journey is 85 km per hour. Her corresponding velocity, however, will be 85 km per hour away from Milton Keynes. A statement of velocity, therefore, must include both the speed and the direction of travel.
The point at which two distancetime graphs cross has a special significance. In this case, each graph represents Alice’s or Bob’s distance from Milton Keynes as time increases. The point at which the graphs cross represents the situation where Alice and Bob are exactly the same distance from Milton Keynes at the same time. The model predicts that, at that moment, they both will be in exactly the same place, although travelling in opposite directions and on opposite sides of the road.
Activity 20: Making predictions
If Alice and Bob follow their plans:

How far apart will they be at 1.30pm?

When and where will they pass each other?
Discussion
At 1.30 pm Alice and Bob will be about 80 km apart.
They will pass at about 265 km north of Milton Keynes, at about 2.10 pm.