Exploring distance time graphs

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# 1.8.2 Distance, speed and time

• Which mathematical formulas are used to relate distance, speed and time?

Look first at distance. If you are travelling at a speed of 30 kilometres per hour, in one hour you will cover a distance of 30 × 1= 30 kilometres. At a speed of 40 kilometres per hour, in two hours you will cover a distance of 40 × 2 = 80 kilometres. And at 50 kilometres per hour, in three hours you will cover a distance of 50 × 3 = 150 kilometres.

• How is distance related to speed and time in general?

The word formula is ‘distance is equal to speed multiplied by time’ or, using some symbols:

distance = speed x time

Instead of writing the words out in full each time, you can use a shorthand to speed things up. In mathematics, single letters are often used to stand for quantities described in words or phrases. A common convention is to use the first letter of the word as a way of remembering what quantity it stands for. In this case, therefore, d can be used to stand for the numerical distance, s can be used to stand for the value of the speed and t can be used to stand for the measure of time. So the formula is written like this:

d = s x t

This formula is a mathematical model of distance, expressed in terms of speed and time. So, if you know what speed you will be travelling at and how long you will be travelling for, you can use the formula to predict how far you will go instead of having to make the actual journey. But the model contains an important assumption.

• Can you see what it is?

The model uses the assumption that the speed is constant over the entire journey. But that is clearly unrealistic; during the course of a journey you slow down, stop and speed up again many times. Your actual speed is not constant but continually changing. More useful, however, is average speed. ‘Average, here, refers to the mean.

• If you travel at an average speed of 55 kilometres per hour (sometimes faster, sometimes slower), then after 1.5 hours how far will you have travelled?

Re-interpreting s as the average value of the speed, the formula predicts:

d = s x t 55 kilometres per hour x 1.5 hours

= 82.5 kilometres

So now you can calculate the distance if you know the average speed and the total time the journey takes. But you can also think of the relationship between distance, average speed and time in another way. Suppose you know the distance between two places and the time it takes to travel between them. How would you calculate the average speed? What formula would you use then?

When you are trying to understand how one quantity is related mathematically to another, it is useful to try one or two calculations with numbers to get a feel for the relationship. Suppose you travelled 40 kilometres and took an hour to do it. What would be your average speed? It would be 40 kilometres an hour. Now suppose that you travelled 60 kilometres in two hours. What would be your average speed then? It would be 60/2 = 30 kilometres per hour.

• So what is the word formula relating the average speed to the distance and the travel time?

The average speed is calculated by dividing the distance travelled by the journey time:

Using the same letters as before the formula is written like this:

Finally, how would you work out the time a journey should take if you knew the distance and the average speed? How long, for example, would it take to travel 90 kilometres at an average speed of 30 kilometres per hour? You would cover the distance in 90/30 = 3 hours.

• So what is the general word formula relating travel time to the distance and the average speed?

The travel time is equal to the distance travelled divided by the average speed:

which you can write as:

So now you have all three forms of the relationship between time, average speed and distance. Here they are again:

Notice that the first formula is the product of time and speed, while the other two have distance divided by time or speed.

## Remembering the formulas

Figure 36shows a way to remember the formulas. Draw a circle with a ‘Y’ in it. Starting at the top of the ‘Y’ just write the letters d, s and t in the spaces in alphabetical order. (It does not matter which way you go round!) Then to find the formula for time simply cover up ‘time’ on the diagram, and you are left with distance over speed. Similarly covering up ‘speed’ gives the formula distance over time. And finally covering up ‘distance’ gives the formula speed multiplied by time.

Figure 36 Remembering the relationship between distance, speed and time

The above is a good example of a mnemonic, that is a device intended to help you remember something. It may or may not actually help. Only you can tell. But realise that there is nothing mathematical to understand about such memory devices. They have been designed to produce the correct result, but there is no conceptual link between Ys in circles and relationships between speed, distance and time. Mnemonics are about remembering, not understanding.

## Activity 15: Finding the right formula

Use the appropriate formula to work out:

1. the average speed in kilometres per hour, if a distance of 25 km is covered in 45 minutes;

2. the distance travelled in kilometres after travelling for 30 minutes at an average speed of 75 kilometres per hour;

3. the time in seconds to cover a distance of 500 metres at an average speed of 10 metres per second.

### Discussion

1. The formula for speed is s=d/t. 45 minutes is equal to 0.75 hours, so the average speed is 25/0.75=33.3 km per hour.

2. The formula for distance is d=s×t. In this case, s is 75 km per hour and t is 30/60=0.5 hours. The distance is 75 x0.5=37.5 km.

3. The formula for travel time is t=d/s. The time to cover 500 metres at 10 metres per second is 500/10=50 seconds.

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