- Interpret the shape and gradient of motion graphs.
- Calculate speed from a distance–time gradient.
- Calculate acceleration from a speed–time gradient.
- Find distance from the area under a speed–time graph.
4.1 Distance–time graphs
On a distance–time graph, the vertical coordinate shows distance from the chosen start and the horizontal coordinate shows time. The gradient is change in distance divided by change in time, so it gives speed. A horizontal line means no change in distance and therefore rest. A straight sloping line means constant speed.
A curve whose gradient becomes steeper represents increasing speed. A curve that becomes less steep represents decreasing speed. Because the vertical quantity is distance rather than displacement, a conventional distance–time graph does not slope downwards; for journeys that return towards the start, a displacement–time graph is required.
4.2 Finding a gradient
For a straight segment, choose two well-separated points on the line and calculate rise/run. For a curve, draw a tangent at the required point and calculate the gradient of the tangent using a large triangle. Do not use a tiny triangle around the point because reading uncertainties then become a large percentage of the changes.
Always use the axis scales and units. If distance is in kilometres and time in minutes, the gradient is initially km min⁻¹ and must be converted if m s⁻¹ is required.
4.3 Speed–time graphs
On a speed–time graph, the vertical coordinate is speed. The gradient is change in speed divided by time, so it gives acceleration. A horizontal line above the time axis is constant speed. A straight rising line is uniform acceleration; a straight falling line is uniform deceleration. A curved line shows changing acceleration.
The area under the graph is speed × time and therefore gives distance. Break a complex region into rectangles, triangles and trapezia. The area under a velocity–time graph gives displacement and areas below the time axis are negative, but for a speed–time graph all areas are positive.
4.4 Linking graphs to descriptions
A complete graph question often asks for a story of the motion. Refer to both the type of motion and the numerical values: “accelerates uniformly from rest to 12 m s⁻¹ in 4 s” is stronger than simply “speeds up”.
Graph axes should occupy more than half the available grid, use a convenient linear scale, include units in headings and show accurately plotted points. A best-fit line or smooth curve should represent the trend rather than joining every point with jagged segments unless instructed.
Worked examples
Distance rises from 20 m at 5 s to 44 m at 8 s. Speed = (44−20)/(8−5) = 8.0 m s⁻¹.
A car accelerates uniformly from 0 to 12 m s⁻¹ in 4 s and then continues at 12 m s⁻¹ for 4 s. Distance = ½×4×12 + 4×12 = 24 + 48 = 72 m.
Speed falls from 12 m s⁻¹ to 3 m s⁻¹ in 4 s. a = (3−12)/4 = −2.25 m s⁻².
Practical focus
Roll a trolley down a ramp and use a motion sensor or light gates. Produce a speed–time graph. Change the ramp angle and compare gradients while keeping the trolley and track conditions constant.
Examination guidance
- State what a gradient or area represents before calculating it.
- Use a large triangle for a graph gradient.
- Do not say a horizontal line on a distance–time graph means constant speed; it means zero speed.
Check your understanding
- What does the area under a speed–time graph represent?
- What feature of a distance–time graph shows speed?
- A speed–time line becomes progressively steeper. What happens to acceleration?
- Distance travelled.
- Its gradient.
- Acceleration increases; it is non-uniform.
