Distance-time and displacement-time graphs - Higher
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 = \( \frac{\text{change in y}}{\text{change in x}} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\text{change in metres}}{\text{change in seconds}} = m/s\)
The speed is \( \frac {20}{10} = 2~m/s\).
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 = \(\frac{\text{change in distance}}{\text{change in time}} = \frac {\text{change in miles}}{\text{change in hours}} = \frac{3}{1}\) = 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).
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