Distance-time and displacement-time graphs - Higher

Distance-time graphs show distance on the vertical axis and time on the horizontal axis.

Graph of time vs distance with triangle of sides 10 and 20 shown along the plot

The gradient of a distance-time graph represents speed because gradient =  \frac{\text{change in y}}{\text{change in x}} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\text{change in metres}}{\text{change in seconds}} = m/s

The speed is  \frac {20}{10} = 2~m/s.

curriculum-key-fact
The gradient of a distance-time graph represents speed.

When displaying a journey, the vertical axis will often represent the distance from a particular place rather than the distance travelled. Such graphs are known as displacement-time graphs.

A distance time graph showing a person's journey. The y axis is labelled 'miles away from home', the x axis is labelled time. A red line shows the various stages of the person's journey

Sections A and C show travelling away from home.

Sections B and D are when the journey has paused for a rest or a wait. Section E shows the return home.

Question

What is the speed in section A?

Speed = gradient = \frac{\text{change in distance}}{\text{change in time}} =  \frac {\text{change in miles}}{\text{change in hours}} = \frac{3}{1} = 3 miles per hour.

Question

What is happening in section B?

They stopped for 30 minutes (0.5 hours) between 9:00 and 9:30.

Question

What is the average speed between 8:00 and 11:00?

Speed = gradient = \frac{\text{change in distance}}{\text{change in time}} =  \frac{\text{change in miles}}{\text{change in hours}}

Average speed =  \frac{8}{3} = 2 \frac{2}{3} miles per hour.

Note: We must use the total change in distance and the total change in time. A common error is to find the average of the speeds for sections A, B and C.

Question

What is the total distance travelled?

16 miles: 8 miles there and 8 miles back.

Question

What is the speed in section E?

Gradient =  \frac {\text{change in distance}}{\text{change in time}} = \frac {\text {change in miles}} {\text{change in hours}} = \frac {-8}{1} = -8

The speed is 8 miles per hour.

Speed or velocity?

The gradient in section E is -8.

The speed is 8 miles per hour because speed is always a positive value. The velocity, however would be -8 miles per hour because the sign indicates direction (with a positive value meaning travelling away from home and negative value meaning travelling towards home).