K326 / K327 / 2027
Kinematics overview

Topic 3 of 6

Distance-time graphs

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.