# Distance-time graphs

## Calculations involving speed, distance and time

The distance travelled by an object moving at constant speed can be calculated using the equation:

distance travelled = speed × time

This is when:

• distance travelled ( ) is measured in metres (m)
• speed ( ) is measured in metres per second (m/s)
• time ( ) is measured in seconds (s)

### Example

A car travels 500 m in 50 s, then 1,500 m in 75 s. Calculate its average speed for the whole journey.

First calculate total distance travelled ( ):

500 + 1,500 = 2,000 m

Then calculate total time taken, :

50 + 75 = 125 s

Then rearrange to find :

= 2,000 ÷ 125

= 16 m/s

If an object moves along a straight line, the distance travelled can be represented by a distance-time graph.

In a distance–time graph, the gradient of the line is equal to the speed of the object. The greater the gradient (and the steeper the line) the faster the object is moving.

### Example

Calculate the speed of the object represented by the green line in the graph, from 0 to 4 s.

change in distance = (8 – 0) = 8 m

change in time = (4 – 0) = 4 s

speed = 8 ÷ 4

speed = 2 m/s

Question

Calculate the speed of the object represented by the purple line in the graph.

change in distance = (10 – 0) = 10 m

change in time = (2 – 0) = 2 s

speed = 20 ÷ 2

speed = 5 m/s

The speed of an object can be calculated from the of a distance-time graph.

## Average speed

Many journeys do not occur at a constant speed. Bodies can speed up and slow down along the journey. However the average speed can still be found for a journey by:

Average speed = total distance travelled ÷ time

### Example

Calculate the average speed of the entire journey of the object following the green line on the graph, from 0 s to 7 s.

Average speed = distance ÷ time

Average speed = 8 ÷ 7

Average speed = 1.14 m/s

## Distance-time graphs for accelerating objects – Higher

If the speed of an object changes, it will be or . This can be shown as a curved line on a distance–time graph.

The table shows what each section of the graph represents:

AIncreasingIncreasing
BConstantConstant
CDecreasingDecreasing
DZeroStationary (at rest)

If an object is accelerating or decelerating, its speed can be calculated at any particular time by:

• drawing a to the curve at that time
• measuring the gradient of the tangent

As the diagram shows, after drawing the tangent, work out the change in distance (A) and the change in time (B).

Note that an object moving at a constant speed is changing direction continually. Since has an associated direction, these objects are also continually changing velocity, and so are accelerating.