K223 / K224 / 2027
Kinematics overview

Lesson 3 of 6

Distance-time graphs

  • I can plot and interpret distance-time and speed-time graphs for motion in one direction.Syllabus

    Syllabus K223 / K224, 2(e). Plot and interpret distance-time and speed-time graphs for motion in one direction.

  • I can use distance-time shapes to identify rest, uniform and non-uniform speed.Syllabus

    Syllabus K223 / K224, 2(f). Use distance-time shapes to identify rest, uniform and non-uniform speed.

Make a guess. You are not marked.

On a graph of distance against time, the line curves upwards, getting steeper as time goes on. What is the object doing?

  1. It is speeding up, because the gradient, its speed, keeps increasing.
  2. It is moving uphill, because the line on the graph goes up.
  3. It is slowing down, because the line is curving away from a straight line.
Show the answer

It is speeding up, because the gradient, its speed, keeps increasing.

The gradient of a distance-time graph is the speed. A line that gets steeper has an increasing gradient, so the object is speeding up.

In short

  • The gradient of a distance-time graph is the speed.
  • A straight sloping line means constant speed. A steeper line means a faster speed.
  • A horizontal line means the object is at rest.

Read distance from the height and speed from the gradient.

Use the definitions of distance and speed. A point such as (2 s, 3 m) pairs a time on the horizontal axis with a reading on the vertical axis.

Time goes on the horizontal axis. Distance travelled from the start goes on the vertical axis. The gradient is the slope of the line: the vertical change divided by the horizontal change. Here it tells you how much distance changes per second.

Speed = gradient = change in distance / change in timeFor a straight segment, choose two well-separated points on that segment.

At rest

Distance / mDistance stays at 2 m. Zero gradient means zero speed.024024Distance / mTime / s
Distance stays at 2 m. Zero gradient means zero speed.

Uniform speed

Distance / mThe straight line rises by 1 m each second. Its constant gradient is 1 m/s.024024Distance / mTime / s
The straight line rises by 1 m each second. Its constant gradient is 1 m/s.

Non-uniform speed

Distance / mThe curve gets steeper. More distance is covered each second, so the object is speeding up.024024Distance / mTime / s
The curve gets steeper. More distance is covered each second, so the object is speeding up.

Worked example

Calculating a gradient

A straight segment joins (2 s, 3 m) to (5 s, 15 m).

Gradient = (15 - 3) / (5 - 2) = 12 / 3 = 4 m/s

The object moves at a constant speed of 4 m/s during this interval. Dividing 15 by 5 would use the origin, which is not on this segment.

A curve has a changing gradient. Where it becomes steeper, the object is moving faster. Where it becomes flatter, it is slowing down. To estimate speed at one instant, draw a tangent that follows the curve's direction at that point, then calculate the tangent's gradient. A line joining two points on the curve gives an average over that interval.

Worked example

Finding the gradient at one instant

Find the speed at 2 s on this curved graph. Draw a straight tangent that follows the curve's direction at the marked point, (2 s, 1 m).

Distance / mA rising curve passes through (2 s, 1 m). The dashed tangent at that point joins A (1 s, 0 m) to B (4 s, 3 m). The gradient triangle has a rise of 3 m and a run of 3 s.0123401234Distance / mTime / sAB
The solid line is the motion graph. The dashed line is its tangent at 2 s. A and B lie on the tangent; they do not have to lie on the curve.
  1. Choose two well-separated points on the tangent: A is (1 s, 0 m) and B is (4 s, 3 m).
  2. Vertical change = 3 - 0 = 3 m. Horizontal change = 4 - 1 = 3 s.
  3. Gradient = 3 m / 3 s = 1 m/s.

The estimated speed is 1 m/s at 2 s. The 3 s interval belongs to the gradient triangle; it does not turn this into an average over the curved motion.

Total distance travelled cannot decrease with time. A flat segment is a stop; a steeper rising segment means greater speed. A downward line cannot represent a decrease in total distance travelled.

A higher line does not necessarily mean faster motion. Compare gradients, not heights. A horizontal line above zero still represents an object at rest.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

A girl walks in a straight line away from her home. A graph of her distance from home against time has three straight parts: from (0 s, 0 m) to (40 s, 60 m), then flat from 40 s to 70 s, then from (70 s, 60 m) to (100 s, 150 m). Describe her motion in each part and calculate her speed in the last part. [3]

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