Distance-time graphs

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

speed = \frac{distance}{time}

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

speed = \frac{10}{2}

speed = 5 m/s

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

Distance-time graphs for accelerating objects

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:

  1. drawing a tangent to the curve at that time
  2. 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).

gradient = \frac{vertical~change}{horizontal~change}

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

A ball orbits around a point, its path is shown by a dotted line, its constant speed is represented by a red arrow and its acceleration is shown by a green arrow pointing to the centre.

The Moon and other satellites orbit the Earth at a constant speed, but are accelerating towards the Earth.