K323 / 2027
Kinematics overview

Topic 3 of 6

Displacement-time graphs

Read displacement from the height and velocity from the gradient.

Use the definitions of displacement and velocity. 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. Displacement from the chosen starting point 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 displacement changes per second.

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

At rest

Displacement / mDisplacement stays at 2 m. Zero gradient means zero velocity.024024Displacement / mTime / s
Displacement stays at 2 m. Zero gradient means zero velocity.

Uniform velocity

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

Non-uniform velocity

Displacement / mThe curve gets steeper. More distance is covered each second, so the object is speeding up.024024Displacement / 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 velocity 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 velocity 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 velocity 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).

Displacement / 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.0123401234Displacement / 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 velocity 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.

A downward-sloping displacement-time line has negative velocity: the object is moving in the chosen negative direction. A negative displacement tells you which side of the origin it is on, not which direction it is moving. Check the gradient for its direction of motion.

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