Velocity-time graphs


Acceleration is the rate of change of velocity. It is the amount that velocity changes per unit time.

The change in velocity can be calculated using the equation:

change in velocity = final velocity – initial velocity

\Delta \text{v} = \text{v - u}

The average acceleration of an object can be calculated using the equation:

\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}}

\text{a} = \frac{\Delta \text{v}}{\text{t}}

This is when:

  • acceleration ( \text{a}) is measured in metres per second squared (m/s2 )
  • change in velocity ( \Delta \text{v}) is measured in metres per second (m/s)
  • time taken ( \text{t}) is measured in seconds (s)

If an object is slowing down, it is decelerating (and its acceleration has a negative value).


A car takes 8.0 s to accelerate from rest to 28 m/s. Calculate the average acceleration of the car.

final velocity, \text{v} = 28 m/s

initial velocity, \text{u} = 0 m/s (because it was at rest – not moving)

change in velocity, \Delta \text{v} = (28 – 0) = 28 m/s

\text{a} = \frac{\Delta \text{v}}{\text{t}}

= 28 ÷ 8

= 3.5 m/s2


A car takes 25 s to accelerate from 20 m/s to 30 m/s. Calculate the acceleration of the car.

final velocity, \text{v} = 30 m/s

initial velocity, \text{u} = 20 m/s

change in velocity, \Delta \text{v} = (30 – 20) = 10 m/s

\text{a} = \frac{\Delta \text{v}}{\text{t}}

= 10 ÷ 25

= 0.4 m/s2

Determining acceleration

If an object moves along a straight line, its motion can be represented by a velocity–time graph. The gradient of the line is equal to the acceleration of the object.

A velocity/time graph. Graph with four distinct sections. All lines are straight.

The table shows what each section of the graph represents:

Section of graphGradientVelocityAcceleration
D (v = 0)ZeroStationary (at rest)Zero

Calculating displacement

Scientists draw graphs of data to help analyse a situation. A velocity-time graph of a journey can give information about acceleration (the gradient) and distance travelled (displacement).

The area under the graph can be calculated by:

  • using geometry (if the lines are straight)
  • counting the squares beneath the line (particularly if the lines are curved)
The displacement of an object can be calculated from the area under a velocity-time graph.


Calculate the total displacement of the object - whose motion is represented by the velocity–time graph below.

The y axis shows velocity in metres per second and the x axis time in seconds.  The object increases its velocity from 0 metres per second to 8 metres per second in 4 seconds.

Here, the displacement can be found by calculating the total area of the shaded sections below the line.

  1. Find the area of the triangle:
    • ½ × base × height
    • ½ × 4 × 8 = 16 m2
  2. Find the area of the rectangle:
    • base × height
    • (10 – 4) × 8 = 48 m2
  3. Add the areas together to find the total displacement:
    • (16 + 48) = 64 m