Distance-time graphs

Distance-time graphs show how far an object has travelled (up the side) against time (along the bottom). The gradient of the line is the speed: a steeper line means faster, a flat line means the object has stopped, and a curve means the speed is changing. For a curve, you find the speed at one moment by drawing a tangent and measuring its gradient.

How to read a distance-time graph

Here x is the time in seconds and y is the distance the runner has travelled in metres. Read the graph section by section:

So the runner runs at a steady speed from the start to A, stops between A and B, then runs faster from B to C, because that section is steeper.

Finding speed from a distance-time graph

The speed is the gradient of the line:

Example 1: the gradient of each section

• Start to A: 200 m in 50 s, so speed = 200 ÷ 50 = 4 m/s.
• A to B: the distance stays at 200 m for 30 s, so the gradient is 0 and the speed is 0 m/s.
• B to C: from 200 m to 400 m in 40 s, so speed = 200 ÷ 40 = 5 m/s.

Always read two points that sit exactly on grid lines, as far apart as possible, and include the units: metres divided by seconds gives m/s.

Example 2: average speed for the whole run

Average speed = total distance ÷ total time = 400 ÷ 120 = 3.3 m/s. Note that this is not the average of 4, 0 and 5: the sections take different amounts of time. See the speed equation guide for more on speed = distance ÷ time.

Velocity-time graphs look similar but mean something different: their gradient is acceleration. Practise reading one in this question from the Laws of Motion course.

Curved distance-time graphs: acceleration

If an object is speeding up, it covers more distance every second, so the line curves upwards and gets steeper. There is no single gradient for a curve, but you can find the speed at one particular time by drawing a tangent.

Here x is the time in seconds and y is the distance in metres, d = 1.5t². The car starts from rest, so the curve starts flat and gets steeper.

Example 3: speed from a tangent

1. Mark the point on the curve at the time you want: (4 s, 24 m).
2. Use a ruler to draw a straight line that just touches the curve at that point, with the same slope as the curve there. This is the tangent.
3. Pick two points far apart on the tangent. For this curve, the tangent at t = 4 s passes through (2 s, 0 m) and (6 s, 48 m).
4. Gradient of the tangent = (48 − 0) ÷ (6 − 2) = 48 ÷ 4 = 12 m/s. That is the car's speed at 4 s.

A curve that gets less steep and then levels off shows an object slowing down and stopping.

Check your understanding.

Drawing a distance-time graph

To draw a graph from measurements, put time on the x-axis and distance on the y-axis, choose scales that use most of the grid, plot each point with a small cross, and join them with straight lines or a smooth curve as the motion suggests. For example, a toy car is timed passing markers:

Every 2 s the car covers another 3 m, so the plotted points form one straight sloping line and the speed never changes: 12 m ÷ 8 s = 1.5 m/s.

Distance-time graphs and displacement-time graphs

Strictly, distance travelled can never go down, so a true distance-time graph never slopes downwards. Many GCSE questions, though, plot the distance from the starting point, and then a downward slope means the object is coming back. A graph of distance in a stated direction is a displacement-time graph, and its gradient is velocity. The displacement-time graph guide explains the difference.

Common mistakes

• Thinking a flat line means constant speed. Flat means stopped; constant speed is a straight sloping line.
• Mixing up distance-time and velocity-time graphs. On a velocity-time graph, a flat line means constant speed.
• Reading the gradient upside down. It is distance ÷ time, rise over run.
• Averaging the speeds of sections instead of dividing total distance by total time.
• Using one point for a gradient. Use the change between two points.
• Drawing a tangent that cuts through the curve. It should only touch the curve at the point.

Practice questions

A cyclist travels 600 m in 120 s at a steady speed, stops for 60 s, then travels another 900 m in 100 s.

1. What is the speed in the first section?
2. What does the graph look like while the cyclist is stopped?
3. What is the speed in the last section, and is that line steeper or less steep than the first?
4. Calculate the average speed for the whole journey.

Answers: 1) 600 ÷ 120 = 5 m/s. 2) A horizontal line at 600 m. 3) 900 ÷ 100 = 9 m/s: steeper. 4) 1500 m ÷ 280 s = 5.4 m/s.

Practise with a short timed paper

Motion graphs come up in GCSE Physics and in 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. For the graph that shows acceleration, see the velocity-time graph guide.

Read next

Physics · MechanicsSpeed equation: speed = distance ÷ time, with examplesThe speed equation, speed = distance ÷ time, and the distance equation s = vt: rearranging, units, km/h to m/s, average speed and distance-time graphs.