Distance–Time Graphs

A distance–time graph shows how the distance of an object from a starting point changes over time. The gradient (slope) of the line gives the speed. A straight line means constant speed, a curved line means changing speed, and a horizontal line means the object is stationary.

What is a Distance–Time Graph?

A distance–time (d–t) graph plots the distance from a reference point (y-axis) against time (x-axis).

The gradient (slope) of a d–t graph gives the speed:

$\text{speed} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\Delta d}{\Delta t}$

Reading the Graph

Line type	Meaning
Horizontal line	Object is stationary (not moving)
Straight line sloping upward	Constant speed moving away
Straight line sloping downward	Constant speed moving back towards start
Curved line (getting steeper)	Accelerating (speeding up)
Curved line (getting shallower)	Decelerating (slowing down)

Note

The steeper the line, the faster the speed (greater gradient = greater speed).

Calculating Speed from a Graph

To find the speed from a straight-line section, pick two points and calculate:

$\text{speed} = \frac{d_2 - d_1}{t_2 - t_1}$

Reading a Distance–Time Graph

A car travels 200 m in 10 seconds (straight line on graph). Find its speed.

$\text{speed} = \frac{200}{10} = 20 \text{ m/s}$

Worked Graph Example

Imagine a person on a walk:

0–5 min: walks steadily away (straight slope)
5–8 min: stops to rest (horizontal)
8–12 min: walks back home at a faster speed (steeper slope, downward)

Real world: Cyclist's journey to school

A cyclist leaves home and the d–t graph shows:

0–5 min: steady uphill climb — shallow, straight slope (slow, constant speed)
5–6 min: stopped at traffic lights — horizontal line (speed = 0)
6–10 min: downhill on a fast road — steeper, straight slope (higher constant speed)

The steeper the slope, the higher the speed. The flat section means a complete stop.

Real world: Bus route with a stop

A bus travels 600 m to a bus stop in 60 s, waits for 30 s, then continues another 400 m in 40 s.

Segment 1 speed: $\dfrac{600}{60} = 10 \text{ m/s}$
Segment 2 (waiting): speed = 0 m/s
Segment 3 speed: $\dfrac{400}{40} = 10 \text{ m/s}$

On the d–t graph, both moving segments have the same gradient — the bus travels at the same speed in both sections.

Real world: Runner accelerating at the start of a race

At the start of a 100 m sprint the runner accelerates from rest. The d–t graph curves upward (getting steeper), showing the runner covers more distance per second as time goes on — a classic sign of acceleration.

Tip

Where do you see this in real life? Train operators and delivery companies use d–t graphs to plan schedules and spot delays. A fitness tracker plots your distance over time throughout a run, showing when you sped up, stopped, or turned around.

Practice Problems

A d–t graph shows a horizontal line. What is the object doing?

Show answer

The object is stationary — it is not moving. The distance from the start stays constant, so speed = 0.

On a d–t graph, between t=0 and t=6s, distance increases from 0 to 30 m. What is the speed?

Show answer

speed = Δd/Δt = 30/6 = 5 m/s

Which section of a d–t graph shows faster movement: a line with gradient 8 m/s or a line with gradient 3 m/s?

Show answer

The line with gradient 8 m/s represents faster movement. A steeper gradient means greater speed.