Distance-time and displacement-time graphs
Distance-time graphs show distance on the vertical axis and time on the horizontal axis.
The gradient of a distance-time graph represents speed because:
gradient = \(\frac{change~in~y}{change~in~x}\) = \( \frac{change~in~distance}{change~in~time} \) = \(\frac{change~in~metres}{change~in~seconds} \) = m/s.
The speed = \(\frac{20}{10} = 2 m/s\)
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.
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?
\(Speed = gradient = \frac{change~in~distance}{change~in~time} = \frac{change~in~miles}{change~in~hours} = \frac{3}{1} =\)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 9:00 and 11:00?
\(Speed = gradient = \frac{change~in~distance}{change~in~time} = \frac{change~in~miles}{change~in~hours}\)
\(Average~speed = \frac{8}{3} = 2 \frac{2}{3}\) 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?
\(gradient = \frac{change~in~distance}{change~in~time} = \frac{change~in~miles}{change~in~hours} = \frac{-8}{1} = -8\)
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).