All subjects · Physics · WPH11 (Unit 1)

Displacement, velocity and acceleration graphs

Motion graphs let you read off velocities, accelerations and distances, even when the acceleration is not constant.

Displacement-time graphs

The gradient of a displacement-time graph is the velocity. A straight line means constant velocity. A curve means the velocity is changing: draw a tangent and find its gradient to get the velocity at that instant.

Velocity-time graphs

For a curved line (non-uniform acceleration), use a tangent for the acceleration and estimate the area by counting squares or splitting it into trapezia.

Acceleration-time graphs

The area under an acceleration-time graph is the change in velocity. A horizontal line means uniform acceleration.

Reading graphs

Always check the axis labels and units before you calculate. Areas below the time axis on a velocity-time graph are displacements in the negative direction.

Worked example

A velocity-time graph rises uniformly from 0 to 12 m/s in 4.0 s, then stays at 12 m/s for 6.0 s. Find the acceleration and the total displacement.

  1. Acceleration = gradient = 12 / 4.0 = 3.0 m/s2
  2. Area of triangle = 0.5 x 4.0 x 12 = 24 m
  3. Area of rectangle = 6.0 x 12 = 72 m

Answer: a = 3.0 m/s2; displacement = 96 m

Key idea: s-t gradient = velocity; v-t gradient = acceleration, v-t area = displacement; a-t area = change in velocity.

Check yourself

  1. What does the gradient of a displacement-time graph give?
    • Acceleration
    • Velocity
    • Displacement
    • Change in velocity
    Show answer

    Velocity. Gradient = change in displacement / change in time = velocity.

  2. What does the area under a velocity-time graph give?
    • Acceleration
    • Velocity
    • Force
    • Displacement
    Show answer

    Displacement. Area = velocity x time = displacement.

  3. On a velocity-time graph, the velocity rises steadily from 2.0 m/s to 8.0 m/s in 3.0 s. Find the acceleration in m/s2.
    Show answer

    2 m/s2. a = (8.0 - 2.0) / 3.0 = 2.0 m/s2.

  4. A car decelerates uniformly from 15 m/s to rest in 6.0 s. Use the area under its velocity-time graph to find the distance travelled, in m.
    Show answer

    45 m. Triangle area = 0.5 x 6.0 x 15 = 45 m.

  5. What does the area under an acceleration-time graph give?
    • Change in velocity
    • Displacement
    • Distance
    • Change in acceleration
    Show answer

    Change in velocity. Area = acceleration x time = change in velocity.

  6. An acceleration-time graph is a horizontal line at 2.0 m/s2 for 5.0 s. The object started at 3.0 m/s. Find its final velocity in m/s.
    Show answer

    13 m/s. Change in velocity = area = 2.0 x 5.0 = 10 m/s, so v = 3.0 + 10 = 13 m/s.

  7. A displacement-time graph curves upwards, getting steeper with time. What is the object doing?
    • Speeding up
    • Slowing down
    • Moving at constant velocity
    • Stationary
    Show answer

    Speeding up. A steeper gradient means a larger velocity, so it is accelerating.

  8. A velocity-time graph is a horizontal line above the time axis. What does this show?
    • Constant acceleration
    • Constant velocity and zero acceleration
    • The object is at rest
    • Increasing acceleration
    Show answer

    Constant velocity and zero acceleration. The velocity does not change, so the gradient (acceleration) is zero.

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