All subjects · Mathematics · WME01 (M1)

Displacement, speed and acceleration-time graphs

Besides velocity-time graphs, the specification lists displacement-time, speed-time and acceleration-time graphs. Each has its own meaning for gradient and area.

Displacement-time

The gradient of a displacement-time graph is the velocity. A horizontal line means at rest; a straight sloping line means constant velocity; a curve means the velocity is changing.

Acceleration-time

The area under an acceleration-time graph is the change in velocity. For constant acceleration the graph is a horizontal line.

Speed-time and vertical motion

Speed is never negative, so a speed-time graph never goes below the axis. For a ball thrown up and caught: the velocity-time graph is one straight line of gradient -9.8 crossing the axis; the speed-time graph is a V shape touching zero at the top. M1 papers often ask you to sketch one of these.

Worked example

A particle starts from rest with acceleration 3 m/s2 for 4 s, then 0 m/s2 for 5 s. Find its speed at t = 9.

  1. Area under the a-t graph from 0 to 4 = 3 × 4 = 12.
  2. No change in velocity from 4 to 9.

Answer: 12 m/s

Key idea: s-t gradient = velocity; v-t gradient = acceleration and area = displacement; a-t area = change in velocity; speed-time never goes negative.

Check yourself

  1. A displacement-time graph is a straight line from (0 s, 4 m) to (5 s, 34 m). Find the velocity.
    Show answer

    6 m/s. Gradient = 30/5 = 6 m/s.

  2. A section of a displacement-time graph is horizontal. What is the particle doing?
    • Moving at constant speed
    • Accelerating
    • Moving backwards
    • At rest
    Show answer

    At rest. Displacement is not changing, so the velocity is zero.

  3. A particle moving at 2 m/s has acceleration 1.5 m/s2 for 6 s. Using an acceleration-time graph, find its final velocity.
    Show answer

    11 m/s. Change in velocity = area = 1.5 × 6 = 9, so v = 2 + 9 = 11 m/s.

  4. A ball is thrown vertically up and caught at the same height. Which describes its speed-time graph?
    • A single straight line crossing the time axis
    • A horizontal line
    • A V shape that touches zero at the highest point
    • A curve that never reaches zero
    Show answer

    A V shape that touches zero at the highest point. Speed falls to zero at the top, then rises again; it never goes negative.

  5. Taking upwards as positive, what is the gradient of the velocity-time graph for a ball in free flight?
    Show answer

    -9.8 m/s2. The acceleration is 9.8 m/s2 downwards, so the gradient is -9.8.

  6. Which feature of a velocity-time graph gives the displacement?
    • The gradient
    • The intercept on the velocity axis
    • The area between the graph and the time axis
    • The length of the line
    Show answer

    The area between the graph and the time axis. Displacement = velocity × time, which is an area.

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Last updated 7 November 2026. Spotted a mistake? Tell us.