K323 / 2027
Kinematics overview

Lesson 3 of 6

Displacement-time graphs

  • I can plot and interpret displacement-time and velocity-time graphs in one dimension.Syllabus

    Syllabus K323, 2(e). Plot and interpret displacement-time and velocity-time graphs in one dimension.

  • I can use displacement-time shapes to identify rest, uniform and non-uniform velocity.Syllabus

    Syllabus K323, 2(f). Use displacement-time shapes to identify rest, uniform and non-uniform velocity.

Make a guess. You are not marked.

On a graph of distance against time, the line curves upwards, getting steeper as time goes on. What is the object doing?

  1. It is speeding up, because the gradient, its speed, keeps increasing.
  2. It is moving uphill, because the line on the graph goes up.
  3. It is slowing down, because the line is curving away from a straight line.
Show the answer

It is speeding up, because the gradient, its speed, keeps increasing.

The gradient of a distance-time graph is the speed. A line that gets steeper has an increasing gradient, so the object is speeding up.

Read displacement from the height and velocity from the gradient.

Use the definitions of displacement and velocity. A point such as (2 s, 3 m) pairs a time on the horizontal axis with a reading on the vertical axis.

Time goes on the horizontal axis. Displacement from the chosen starting point goes on the vertical axis. The gradient is the slope of the line: the vertical change divided by the horizontal change. Here it tells you how much displacement changes per second.

Velocity = gradient = change in displacement / change in timeFor a straight segment, choose two well-separated points on that segment.

At rest

Displacement / mDisplacement stays at 2 m. Zero gradient means zero velocity.024024Displacement / mTime / s
Displacement stays at 2 m. Zero gradient means zero velocity.

Uniform velocity

Displacement / mThe straight line rises by 1 m each second. Its constant gradient is 1 m/s.024024Displacement / mTime / s
The straight line rises by 1 m each second. Its constant gradient is 1 m/s.

Non-uniform velocity

Displacement / mThe curve gets steeper. More distance is covered each second, so the object is speeding up.024024Displacement / mTime / s
The curve gets steeper. More distance is covered each second, so the object is speeding up.

Worked example

Calculating a gradient

A straight segment joins (2 s, 3 m) to (5 s, 15 m).

Gradient = (15 - 3) / (5 - 2) = 12 / 3 = 4 m/s

The object moves at a constant velocity of +4 m/s during this interval. Dividing 15 by 5 would use the origin, which is not on this segment.

A curve has a changing gradient. Where it becomes steeper, the object is moving faster. Where it becomes flatter, it is slowing down. To estimate velocity at one instant, draw a tangent that follows the curve's direction at that point, then calculate the tangent's gradient. A line joining two points on the curve gives an average over that interval.

Worked example

Finding the gradient at one instant

Find the velocity at 2 s on this curved graph. Draw a straight tangent that follows the curve's direction at the marked point, (2 s, 1 m).

Displacement / mA rising curve passes through (2 s, 1 m). The dashed tangent at that point joins A (1 s, 0 m) to B (4 s, 3 m). The gradient triangle has a rise of 3 m and a run of 3 s.0123401234Displacement / mTime / sAB
The solid line is the motion graph. The dashed line is its tangent at 2 s. A and B lie on the tangent; they do not have to lie on the curve.
  1. Choose two well-separated points on the tangent: A is (1 s, 0 m) and B is (4 s, 3 m).
  2. Vertical change = 3 - 0 = 3 m. Horizontal change = 4 - 1 = 3 s.
  3. Gradient = 3 m / 3 s = 1 m/s.

The estimated velocity is +1 m/s at 2 s. The 3 s interval belongs to the gradient triangle; it does not turn this into an average over the curved motion.

A downward-sloping displacement-time line has negative velocity: the object is moving in the chosen negative direction. A negative displacement tells you which side of the origin it is on, not which direction it is moving. Check the gradient for its direction of motion.

A higher line does not necessarily mean faster motion. Compare gradients, not heights. A horizontal line above zero still represents an object at rest.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

A girl walks in a straight line away from her home. A graph of her distance from home against time has three straight parts: from (0 s, 0 m) to (40 s, 60 m), then flat from 40 s to 70 s, then from (70 s, 60 m) to (100 s, 150 m). Describe her motion in each part and calculate her speed in the last part. [3]

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