Distance-time graphs
A distance-time graph shows how the distance travelled by an object changes over time and can be used to calculate the speed of the object.
A horizontal line on a distance-time graph shows that the object is Not moving or at rest., or not moving, because the distance does not change as time changes.
A sloping line on a distance-time graph shows that the object is moving because the distance it travels increases as time increases.
To calculate the gradient of a straight-line graph:
- Choose two points on the line that are far apart.
- Using a ruler, draw a right-angled triangle from one point to the other.
- Use the y-axis scale to work out the height or rise of the triangle.
- Use the x-axis scale to work out the width or run of the triangle.
Gradient = \(\frac{change~in~y}{change~in~x}\).
But this is often easier to remember as:
Gradient = \(\frac{rise}{run}\).
Example: Using the gradient or slope of a distance - time graph to work out the speed of an object
Calculate the speed of the object represented by the green line in the graph below, from 0 to 3 s.
Answer
A right-angled triangle is drawn using two far apart points – in this case using the points on the green line at 0 seconds and 3 seconds.
Rise = 6 m.
Run = 3 s.
Gradient = \(\frac{rise}{run}\).
Gradient = \(\frac{6}{3}\).
Gradient = 2 m/s.
The speed of the object represented by the green line in the graph is 2 m/s.
Question
Calculate the average speed of the object represented by the purple line in the graph, from 0 to 2 s.
Draw a right-angled triangle between the points on the purple line at 0 seconds and 2 seconds.
Rise = 10 m.
Run = 2 s.
Gradient = \(\frac{rise}{run}\).
Gradient = \(\frac{10}{2}\).
Gradient = 5 m/s.
Speed = gradient of distance-time graph = 5 m/s.
The speed of the object represented by the purple line in the graph is 5 m/s.
Question
Calculate the average speed of the object represented by the green line in the graph, from 0 to 10 s.
The slope of the green line changes over the 10 seconds and so to work out the average speed use:
Average speed = \(\frac{distance}{time}\).
Distance = 7 m.
Time = 10 s.
Average speed = \(\frac{7}{10}\).
Average speed = 0.7 m/s.
The average speed of the object represented by the green line, from 0 to 10 s is 0.7 m/s.
Look at this distance-time graph for a car travelling along a road and answer the following questions.
Question
How far did the car travel in the first 4 seconds?
Go to 4 seconds on the x-axis.
Go vertically up until you meet the line of the graph.
Then trace across horizontally until you meet the y-axis.
The reading on the y-axis equals the distance travelled by the car in the first 4 s.
The distance travelled by the car in the first 4 seconds = 30 m.
Question
What was the speed of the car over the first 4 seconds?
Draw a right-angled triangle between the points on the line at 0 seconds and 4 seconds.
Rise = 30 m.
Run = 4 s.
Gradient = \(\frac{rise}{run}\).
Gradient = \(\frac{30}{4}\).
Gradient = 7.5 m/s.
Speed = gradient of distance-time graph = 7. 5 m/s.
The speed of the car over the first 4 seconds = 7. 5 m/s.
Question
How long was the car stationary?
The distance moved remains at 30 m between 4 and 8 seconds.
The car was stationary for a total of 4 seconds.
Question
What was the average speed of the car over the entire journey?
The slope of the line changes over the 10 seconds and so to work out the average speed for the entire journey use:
Average speed = \(\frac{distance}{time}\).
Distance = 40 m.
Time = 10 s.
Average speed = \(\frac{40}{10}\).
Average speed = 4 m/s.
The average speed of the car over the entire journey = 4 m/s.
