Velocity-time graphs

A velocity-time graph shows how an object's velocity changes over time, with time on the horizontal axis and velocity on the vertical axis. Two facts unlock every question: the gradient of the line is the acceleration, and the area under the line is the distance travelled (strictly, the displacement). A flat line means constant velocity, and a sloping line means the object is speeding up or slowing down.

Reading a velocity-time graph

Here is a car that pulls away from traffic lights, cruises, then brakes to a stop:

Here x is time in seconds and y is velocity in m/s. The graph has three stages:

Gradient of a velocity-time graph: acceleration

Rise over run on this graph means velocity gained per second, which is exactly what acceleration measures:

Example 1: acceleration of each stage

• 0 to 10 s: a = (20 − 0) ÷ 10 = 2 m/s².
• 10 to 30 s: the velocity does not change, so a = 0 m/s².
• 30 to 40 s: a = (0 − 20) ÷ 10 = −2 m/s². The minus sign means it is slowing down: a deceleration of 2 m/s².

Try reading one yourself in the Laws of Motion course, then press Mark.

Area under a velocity-time graph: distance

Velocity × time = distance, and on the graph velocity × time is an area. So the area under the line, down to the time axis, is the distance travelled. Split the shape into rectangles and triangles, or use the trapezium (trapezoid) area formula.

Example 2: total distance for the car

• 0 to 10 s, triangle: ½ × 10 × 20 = 100 m.
• 10 to 30 s, rectangle: 20 × 20 = 400 m.
• 30 to 40 s, triangle: ½ × 10 × 20 = 100 m.

Total distance = 100 + 400 + 100 = 600 m. You could also treat the whole shape as one trapezium with parallel sides 40 s and 20 s and height 20 m/s: ½ × (40 + 20) × 20 = 600 m.

Example 3: average speed

The SUVAT equations are these same gradient and area facts written as formulas, for any stage with constant acceleration.

Curved velocity-time graphs

If the line is curved, the acceleration is changing. You can still find the gradient and the area, but you have to estimate them.

Here x is time in seconds and y is velocity in m/s. The curve gets steeper, so the object's acceleration is increasing.

Example 4: estimating the area with strips

Split the area from 0 to 6 s into three strips, each 2 s wide, and treat each as a trapezium. The velocities at 0, 2, 4 and 6 s are 0, 2, 8 and 18 m/s:

The estimate is 38 m. It is a slight overestimate, because the straight tops of the trapezia sit just above the curve. The exact answer, found with integration at A-level, is 36 m. More, narrower strips give a better estimate. On squared paper, counting squares works too: find the distance one square stands for, then count the squares under the line.

Example 5: estimating acceleration with a tangent

To find the acceleration at one moment on a curve, draw a tangent: a straight line that just touches the curve at that point. Its gradient is the acceleration at that instant. For this curve, the tangent at t = 4 s goes through (4, 8) and (6, 16), so the acceleration is (16 − 8) ÷ (6 − 4) = 4 m/s².

Check your understanding.

Velocity or speed?

Velocity has a direction, so it can be negative: a line below the time axis means the object is moving backwards. Area below the axis counts as negative displacement. If a question asks for total distance, add the sizes of the areas above and below the axis. If it asks for displacement, subtract the area below from the area above. A speed-time graph never goes below the axis, because speed has no direction.

Common mistakes

• Mixing up velocity-time and distance-time graphs. The same shape means different things on each.
• Reading the gradient as distance or the area as acceleration. Gradient → acceleration, area → distance.
• Forgetting the ½ for triangle areas.
• Unit slips. If time is in minutes or velocity in km/h, convert before working out the area, or the distance will be in strange units.
• Dropping the minus sign on a deceleration. A downward slope gives a negative gradient.
• Calling a flat line "stopped". On a velocity-time graph it means constant velocity.

Practise with a short timed paper

Motion graphs come up in GCSE Physics and GCSE Maths Higher. The daily GCSE Physics Higher and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day.

Read next

Maths · MechanicsDisplacement-time graph: gradient, velocity and examplesHow to read a displacement-time graph: gradient as velocity, flat, steep and curved lines, tangents, and how it differs from a distance-time graph.