Physics

2.5 - Distance-Time Graphs

8 min

Gradient and speed

Stationary and changing speed

Speed from a tangent

Summary

note 1 of 5

Reading a distance-time graph

A distance-time graph shows how far an object has travelled from its starting point over time.
A flat section means the object is stationary (at rest) - the distance doesn't change.
A straight, sloping section means the object is moving at a steady speed - the steeper the line, the faster it's going.
A curve means the speed is changing (accelerating or decelerating):
- A curve that gets steeper means the object is speeding up.
- A curve that levels off means the object is slowing down.

A distance-time graph whose line rises straight, flattens, then curves upward and levels off, the four parts labelled steady speed, stationary, acceleration and deceleration, with axes distance in metres and time in seconds.

note 2 of 5

Finding speed from the gradient

The gradient (slope) of a distance-time graph at any point equals the speed of the object at that point.
For non-uniform speed, the average speed is the total distance travelled divided by the total time taken.

Formula for speed

$Speed = gradient = \frac{change in distance}{change in time}$

note 3 of 5

Worked example - Speed from a straight section

A runner moves in a straight line. On a distance-time graph, a straight section of the line rises from 0 m to 40 m between 0 s and 5 s. Calculate the runner's speed during this section.

A distance-time graph with one straight green line rising from the origin to 40 m at 5 s, distance in metres up the axis and time in seconds along it.

Step 1 - Read the change in distance and change in time

Change in distance = 40 m - 0 m = 40 m

Change in time = 5 s - 0 s = 5 s

Step 2 - Find the gradient

$Speed = \frac{40}{5} = 8 m/s$

The runner's speed during this section is 8 m/s.

note 4 of 5

Speed on a curve from a tangent

When a distance-time graph is curved, the speed is changing, so the gradient is different at every point.
To find the speed at a particular moment, draw a tangent to the curve at that point.
A tangent is a straight line that just touches the curve and runs parallel to it at that point.
The speed at that moment is the gradient of the tangent, worked out from the change in distance (Δs) and the change in time (Δt) marked along it:

$Speed = gradient of tangent = \frac{change in distance}{change in time}$

A distance-time graph with a green curve that gets steeper and an orange tangent touching it near 3 s, dashed lines under the tangent forming a triangle labelled Δs upright and Δt across.

note 5 of 5

Worked example - Speed on a curve from a tangent

A cyclist accelerates, so the distance-time graph of the journey is a curve. To find the speed at 10 s, a tangent is drawn to the curve at that point. The tangent rises 24 m over a horizontal interval of 4 s. Calculate the speed at 10 s.

A distance-time graph running to 15 s with a green curve that steepens, an orange tangent touching it at about 10 s, and dashed lines marking a triangle labelled Δs and Δt.

Step 1 - Read the change in distance and change in time along the tangent

Change in distance = 24 m

Change in time = 4 s

Step 2 - Find the gradient of the tangent

$Speed = \frac{24}{4} = 6 m/s$

The cyclist's speed at 10 s is 6 m/s.

key terms

TermDefinition

TangentA straight line that touches a curve at a single point, used to find the gradient of the curve at that point.

end of notes

Key points to remember

Reading a distance-time graph

A distance-time graph shows how far an object has travelled from its starting point over time.

A flat section means the object is stationary (at rest) - the distance doesn't change.

A straight, sloping section means the object is moving at a steady speed - the steeper the line, the faster it's going.

A curve means the speed is changing (accelerating or decelerating):

A curve that gets steeper means the object is speeding up.
A curve that levels off means the object is slowing down.

Speed on a curve from a tangent

When a distance-time graph is curved, the speed is changing, so the gradient is different at every point.

To find the speed at a particular moment, draw a tangent to the curve at that point.

A tangent is a straight line that just touches the curve and runs parallel to it at that point.

The speed at that moment is the gradient of the tangent, worked out from the change in distance (Δs) and the change in time (Δt) marked along it:

Speed = gradient of tangent = \frac{change in distance}{change in time}