All subjects · Mathematics · WME01 (M1)

Velocity-time graphs

A velocity-time graph shows a whole journey at a glance. M1 often gives a journey in words, asks you to sketch the graph, then uses its area and gradients to find unknowns.

Reading the graph

Trapezium journeys

A journey that accelerates, cruises and decelerates makes a trapezium. Area = (1/2)(sum of parallel sides) × height = (1/2)(total time + cruising time) × top speed. Form an equation in the unknown and solve.

Negative velocity

Area below the time axis means motion in the negative direction. Displacement = area above minus area below; distance travelled = area above plus area below.

Worked example

A train accelerates uniformly from rest for 20 s to V m/s, travels at V m/s for 60 s, then decelerates uniformly to rest in 30 s. It travels 2550 m. Find V.

  1. Total time = 110 s, cruise time = 60 s.
  2. Area = (1/2)(110 + 60)V = 85V.
  3. 85V = 2550.

Answer: V = 30 m/s

Key idea: Gradient = acceleration; area = displacement. Trapezium area = (1/2)(a + b)h. Below the axis counts as negative displacement.

Check yourself

  1. A car accelerates from rest to V m/s in 10 s, stays at V for 40 s, then decelerates to rest in 20 s, travelling 1650 m in total. Find V.
    Show answer

    30 m/s. Area = (1/2)(70 + 40)V = 55V = 1650, so V = 30 m/s.

  2. For the train in the worked example, find the acceleration in the first 20 s.
    Show answer

    1.5 m/s2. Gradient = 30/20 = 1.5 m/s2.

  3. For the train in the worked example, find the distance travelled while decelerating.
    Show answer

    450 m. Triangle area = (1/2)(30)(30) = 450 m.

  4. A bus starts from rest, accelerates at 1 m/s2 for 12 s, travels at constant speed for T s, then decelerates at 2 m/s2 to rest. It travels 360 m. Find T.
    Show answer

    21 s. Top speed 12 m/s. Accelerating: 72 m. Decelerating: takes 6 s, covers 36 m. So 12T = 360 - 72 - 36 = 252 and T = 21 s.

  5. A particle's velocity decreases uniformly from 8 m/s to -8 m/s over 4 s. Find the total distance it travels in these 4 s.
    Show answer

    16 m. It reaches 0 at t = 2. Each triangle has area (1/2)(2)(8) = 8, so the distance is 16 m (the displacement is 0).

  6. On a velocity-time graph, what does a straight line sloping downwards but staying above the time axis show?
    • The particle is moving backwards
    • The particle is slowing down while still moving forwards
    • The particle is at rest
    • The particle has constant velocity
    Show answer

    The particle is slowing down while still moving forwards. Velocity is positive (moving forwards) and decreasing (negative gradient).

Practise with instant feedback

Sources

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