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

\text{s} = \text{vt}

This is when:

  • distance travelled ( \text{s}) is measured in metres (m)
  • speed ( \text{v}) is measured in metres per second (m/s)
  • time ( \text{t}) 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 ( \text{s}):

500 + 1,500 = 2,000 m

Then calculate total time taken, \text{t}:

50 + 75 = 125 s

Then rearrange \text{s} = \text{vt} to find \text{v}:

\text{v} = \frac{\text{s}}{\text{t}}

\text{t} = 2,000 ÷ 125

\text{t} = 16 m/s

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

curriculum-key-fact
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.
A distance time graph shows distance travelled measured by time.

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

\text{speed} = \frac{\text{distance}}{\text{time}}

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

\text{speed} = \frac{\text{distance}}{\text{time}}

speed = 20 ÷ 2

speed = 5 m/s

curriculum-key-fact
The speed of an object can be calculated from the gradient 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 accelerating or decelerating. This can be shown as a curved line on a distance–time graph.

A graph to show distance travelled by time. A shows acceleration, B shows constant speed, C shows deceleration, and A shows stationary position. Three dotted lines separate each section.

The table shows what each section of the graph represents:

Section of graphGradientSpeed
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 tangent to the curve at that time
  • measuring the gradient of the tangent
A distance x time graph, showing a tangent on a curve.

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

\text{Gradient} = \frac{\text{vertical change (A)}}{\text{horizontal change (B)}}

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