Displacement-time graphs

A displacement-time graph shows how far an object is from its starting point, in a given direction, as time goes on. Displacement goes up the side and time along the bottom. The gradient of the graph is the velocity: a steep line means fast, a flat line means stationary, a line sloping down means moving back the other way, and a curve means the velocity is changing.

How to read a displacement-time graph

A walker sets off from home along a straight path, stops, then walks back past home and returns. Positive displacement means on one side of home and negative means on the other.

Here x is the time in seconds and y is the displacement from home in metres. Each part of the line tells you something:

• Straight line: constant velocity.
• Steeper line: greater speed.
• Flat (horizontal) line: stationary.
• Line sloping down: moving in the negative direction, back towards the start or beyond it.
• Curve: the velocity is changing, so the object is accelerating or decelerating.

Gradient of a displacement-time graph: velocity

Velocity tells you how fast displacement changes, so it is the gradient of the graph:

Example 1: the velocity of each section

The minus sign on B to C means the walker is moving in the opposite direction, at a speed of 4 m/s. Speed is never negative; velocity can be, because it has a direction.

Displacement-time vs distance-time graphs

Distance does not involve direction, while displacement is the straight-line distance from the start in a given direction. That changes the graph whenever an object turns back:

Example 2: distance travelled and displacement

For the walk, the distance travelled is 20 + 0 + 40 + 20 = 80 m, but the final displacement is 0 m, because the walker ends where they began. So:

A distance-time graph of the same walk would rise from 0 to 80 m and never come down. GCSE Physics uses distance-time graphs, where the gradient is speed; A-level and Maths mechanics use displacement-time graphs.

Curved displacement-time graphs

When the velocity changes, the graph curves. Getting steeper means speeding up; getting flatter means slowing down. The velocity at one instant is the gradient of the tangent: a straight line that just touches the curve at that point.

Here x is the time in seconds and y is the displacement in metres, with s = t². The curve gets steeper, so the car is accelerating.

Example 3: velocity from a tangent

Find the velocity of the car at t = 3 s.

1. Draw the tangent at (3, 9). For this curve it crosses the time axis at t = 1.5 s and reaches 21 m at t = 5 s.
2. Gradient = (21 − 0) ÷ (5 − 1.5) = 21 ÷ 3.5 = 6 m/s.

A hand-drawn tangent gives an estimate, so exam mark schemes accept a range. Use two points far apart on your tangent to make the gradient more accurate. Drawing a tangent to find the speed is Higher tier only in AQA GCSE Physics.

Example 4: a ball thrown straight up

A ball is thrown upwards at 20 m/s. Taking up as positive and g as 10 m/s², its displacement is s = 20t − 5t²:

The graph is an upside-down parabola. It is steep at the start, flat at the top at t = 2 s, where the velocity is zero for an instant, and steep downwards as the ball falls back. Sketch it, and you will see the gradient change smoothly from positive, through zero, to negative.

Check your understanding.

From a displacement-time graph to a velocity-time graph

Each straight section of a displacement-time graph becomes a flat section of a velocity-time graph, at the height of its gradient. For the walk: 2 m/s for 10 s, 0 for 15 s, −4 m/s for 10 s, then 2 m/s for 10 s. The area under that velocity-time graph gives back the displacement: 20 + 0 − 40 + 20 = 0 m. See the velocity-time graph guide for gradients and areas there. Then try reading one in this question from the Laws of Motion course.

Common mistakes

• Reading the gradient as acceleration. On a displacement-time graph the gradient is velocity; acceleration is the gradient of a velocity-time graph.
• Thinking a flat line means constant velocity. Flat means stationary. A straight sloping line means constant velocity.
• Dropping the minus sign when the line slopes down. It shows the direction.
• Adding displacements to get distance. For distance, add the size of every movement, ignoring direction.
• Using a chord instead of a tangent on a curve. A line joining two points of the curve gives an average velocity, not the velocity at an instant.
• Misreading the scale. Check what each square is worth on both axes before working out a gradient.

Practise with a short timed paper

Motion graphs come up in GCSE Physics and GCSE Maths. The daily GCSE Physics Higher and GCSE Maths Higher papers are short timed papers of 8 questions, with a new one every day. To check a motion problem, use the SUVAT Calculator, and see SUVAT equations for constant acceleration.

Read next

Maths · MechanicsVelocity-time graph: gradient, area and examplesHow to read a velocity-time graph: the gradient gives acceleration and the area gives distance, with GCSE examples, curved graphs and common mistakes.