1.8.7 Distance-time graphs: a mathematical story
Distance-time graphs are a means of replacing a description given in words by a mathematical description of the same event. What follows is a narrative account: that is, a description in the form of story about a bicycle ride. Read the story and then think about how you would use this account to produce a mathematical model of the ride in the form of a distance-time graph.
Sunday started a bit cloudy. The temperature was about 13°C, but I thought I’d keep to the original plan and go cycling with the kids on the track around the local reservoir. The eight-year-old has got his own bike; I can hire a bike for me with the four-year-old on a seat on the back when we get there. What we usually do is to cycle from the bike-hire place to a pub where we can sit outside and have some lunch – burger and chips probably. The pub’s not far – about 5 km from the start and it takes us about 25 minutes to get there. We usually stop for about 45 minutes. After lunch, we go on another 2 kilometres, stop for about 15 minutes and then head back the way we came. I guess our average cycling speed between stops is about 10 km per hour. On the way back, we usually stop at a playground for half an hour. From there it is 3 km to the bike-hire shop which takes us about 15 minutes.
First of all, notice that this narrative account contains a mix of information. There is speed and time and distance information to be sure, but there are also other items which you will not be able to fit easily into a mathematical description, such as the comments about the weather, the location of the cycleway, the ages of the children, and what was for lunch. All these details are important from a personal point of view; but as far as the mathematical model of the bicycle ride is concerned they have no bearing whatsoever. Mathematics, therefore, is not an alternative language; it has no means of speaking of many events that people find important. What it does offer in this example, however, is a way of revealing and representing very specific features of the journey that are embedded in the narrative account.
To build a mathematical model of the bicycle ride you need to be selective about the information you choose. You may also have to piece information together and make some assumptions. Narrative accounts are not mathematical accounts and there may be some inconsistencies you have to resolve before you can put together a reasonable mathematical story.
Activity 17: A mathematical story
Use the information in the story and the relationships between distance, average speed and time to complete Table 4.
Table 4 Time and distances for the cycle ride
|Time from start (minutes)||Distance from start (km) along the route||Comments|
|Arrive at bike-hire|
The distance travelled after lunch is 2 km at 10 km per hour, which takes 0.2 hour, or 12 minutes. The total distance from the start is 7 km.
The playground stop is 3 km from the cycle hire shop, and hence 4 km back from the furthest point on the ride. At an average speed of 10 km per hour, it will take 0.4 hours, or 24 minutes to reach the playground.
Table 5 shows all the distances and times for the ride.
Table 5 Completed distance and time data
|Time from start (minutes)||Distance from start (km)||Comments|
|0||0||leave cycle-hire shop|
|166||0||arrive at cycle-hire shop|
A partially-drawn distance-time graph of the cycle ride is shown in Figure 46. The first straight-line section represents the outward 5 km journey to the pub, which takes 25 minutes. The average speed is represented by the gradient of the graph, which is:
This is equivalent to 0.2 × 60 = 12 km per hour.
During the lunch stop, the distance from the start does not change. Obviously the speed is zero, so the graph is a straight line with a slope of zero. The length of the line represents the length of time (45 minutes) spent at the pub. After lunch, the journey continues for 2 km at an average speed of 10 km per hour. This part of the journey therefore takes 2/10 = 0.2 hours (remember that travel time is equal to distance travelled divided by average speed), or 12 minutes. After this short ride comes the 15 minute break. Once again, the distance from the start does not change over this time, and so this section of the distance-time graph is a horizontal line too.
Activity 18: Completing the graph
Complete the distance-time graph in Figure 46 for the journey back to the cycle hire shop. What is the average speed for this part of the ride?
The distance-time graph for the complete cycle ride is shown in Figure 47. The 7 km journey back takes 166 – 97=69 minutes (including the stop at the playground). So the average speed is (7/69)×60=6.1 km per hour. This speed is represented by the slope of the dashed line on the graph. The line has a negative gradient, indicating that the direction of the journey is back towards the starting point.