Distance–Time Graph (Speed)

Statement

On a distance–time graph, the speed of an object is represented by the gradient (slope) of the line. The formula is:

\[ \text{Speed} = \text{gradient} = \frac{\Delta \text{distance}}{\Delta \text{time}} \]

In other words, speed equals the change in distance divided by the change in time.

Why it’s true

• A distance–time graph plots distance travelled against time.
• The steepness (gradient) of the graph tells us how fast distance increases with time.
• The formula for a gradient is "rise over run", which here is "distance over time".
• Therefore, the gradient of the line equals the speed.

Recipe (how to use it)

1. Pick two points on the straight-line section of the distance–time graph.
2. Find the change in distance (\(\Delta d\)) and change in time (\(\Delta t\)).
3. Divide distance by time: \( \text{Speed} = \Delta d / \Delta t \).
4. If the graph has curves, draw a tangent at the point and find its gradient to get instantaneous speed.

Spotting it

This formula applies whenever you are interpreting a distance–time graph, whether for constant speed (straight line) or variable speed (curved line).

Common pairings

• Uniform motion problems (straight-line graph).
• Acceleration problems (where gradient changes).
• Comparing speeds of two objects by comparing gradients.

Mini examples

1. Given: In 2 hours, a car travels 100 km. On the graph, line joins (0,0) to (2,100). Answer: Gradient = 100/2 = 50 km/h.
2. Given: A train covers 60 km in 1.5 h. Answer: Gradient = 60/1.5 = 40 km/h.

Pitfalls

• Mixing up axes: time should always be on the horizontal axis.
• Using total distance instead of change in distance.
• Forgetting to keep units consistent (minutes vs hours).

Exam strategy

• Clearly label the two points used to calculate the gradient.
• Show working with rise/run to avoid errors.
• If graph is curved, draw tangent carefully.

Summary

The gradient of a distance–time graph gives the speed. A steeper line means a higher speed. This method works for both constant and changing speeds, making it a powerful tool in interpreting motion graphs.