5.13 Distance-time graphs

Distance-time graphs

A distance-time graph shows how far an object has moved from its starting position over time. The most important rule is:

\text{speed} = \text{gradient of a distance-time graph}

The gradient is calculated by dividing the change in distance by the change in time.

Reading the graph

The graph above shows a journey in four stages.

• A to B (0-10 s): a straight sloping line. The distance increases by the same amount each second, so the object moves at a constant speed. The speed is the gradient: \frac{20 \text{ m}}{10 \text{ s}} = 2.0 \text{ m/s}.
• B to C (10-20 s): a horizontal line. The distance does not change, so the object is stationary (speed = 0 m/s).
• C to D (20-30 s): a steeper straight line. The object moves at a larger constant speed: \frac{40 \text{ m}}{10 \text{ s}} = 4.0 \text{ m/s}.
• D to E (30-40 s): a line sloping downwards. The distance from the start is decreasing, so the object is returning to its starting point. The speed is \frac{60 \text{ m}}{10 \text{ s}} = 6.0 \text{ m/s} (the line goes down, so the velocity is in the opposite direction).

Rules for reading distance-time graphs

Calculating speed from the gradient

To find the speed from a graph, pick two points on the straight section of the line, far apart so that the reading is more accurate.

\text{speed} = \frac{\text{change in distance}}{\text{change in time}} = \frac{d_2 - d_1}{t_2 - t_1}

Worked example. On the graph, between 20 s and 30 s the distance goes from 20 m to 60 m.

\text{speed} = \frac{60 - 20}{30 - 20} = \frac{40}{10} = 4.0 \text{ m/s}

Curved lines and tangents (Higher tier)

When an object is accelerating, the distance-time graph is a curve, because the gradient (the speed) is changing all the time. To find the speed at one particular time, you draw a tangent to the curve and calculate its gradient.

1. Choose the time you are interested in, and find the point on the curve.
2. Use a ruler to draw a straight line that touches the curve at that point only, and has the same slope as the curve there.
3. Pick two points on the tangent far apart, and calculate the gradient.

In the example, the tangent at 4 s passes through (2 s, 0 m) and (6 s, 16 m):

\text{speed at 4 s} = \frac{16 - 0}{6 - 2} = \frac{16}{4} = 4.0 \text{ m/s}

Common mistakes

• A horizontal line on a distance-time graph means the object is stopped. On a velocity-time graph it means constant velocity, which is different. Always look at the axes first.
• A line sloping downwards does not mean "slowing down". It means that the object is getting closer to the start.
• The steepness matters, not the height. A steep line at a low value is a faster object than a shallow line at a high value.