12.04 Distance time graphs
Distance-time graphs are a way to describe the movement of people or objects. They usually describe a trip that leaves and returns to a point (like a home base).
The vertical axis of a distance-time graph is the distance travelled from a starting point and the horizontal axis is the time taken from the starting point.
Reading a distance-time graph
There are certain important features of a distance-time graph that we can use to interpret the journey being described:
- As the line moves away from the horizontal axis, the object is moving further away from the "home" point
- As the line moves back towards the horizontal axis, the object is returning home
- When the line is horizontal, the object is not moving
- The steeper the line, the greater the speed of the an object (the faster it moves)
- A straight line indicates a steady speed
- The total distance of the trip is the distance away from and returning home
Worked example
Consider the following graph which displays a day long car tip with the horizontal axis being time in hours and the vertical axis being distance from home in kilometres:
(a) What speed did the car travel in the first hour?
Think: We know that $\text{Speed}=\frac{\text{Distance}}{\text{Time}}$Speed=DistanceTime. How far did they travel in the first hour?
Do:
| Speed | $=$= | $\frac{\text{Distance}}{\text{Time}}$DistanceTime |
| $=$= | $\frac{80\text{ km}}{1\text{ h}}$80 km1 h | |
| $=$= | $80$80 km/h |
(b) What happened between the times of $1$1 and $2$2?
Think: What does it mean for the graph to be horizontal?
Do: The car was stationary for $1$1 hour. Perhaps a break for lunch or a visit to a park.
(c) How far is the car from home after $6$6 hours?
Think: Read the vertical axis for the distance at $6$6 hours.
Do: The car is $50$50 km from home.
(d) What was the average speed of the car over the $6$6 hour journey?
Think: How far has the car travelled in total? The car initially travelled $80$80 km, then was stationary for one hour, then travelled a further $120$120 km before starting the return tip home at $4$4 hours into the journey. In the last section of the journey they are returning to home from $200$200 kilometres away and reach $50$50 kilometres from home, thus they travel $150$150 km.
Do:
| Total distance travelled | $=$= | $80+120+150$80+120+150 km |
| $=$= | $350$350 km |
| Average speed | $=$= | $\frac{\text{Total distance}}{\text{time}}$Total distancetime |
| $=$= | $\frac{350\ km}{6\ h}$350 km6 h | |
| $=$= | $58.\overline{3}$58.3 km/h |
Practice questions
Question 1
Which graph shows the height of a ball being thrown vertically into the air?
A
B
C
D
Question 2
Ben travels forwards and backwards along a straight line.
The graph shows Ben's distance from his starting point at various times of the day.
A line graph is displayed with the horizontal axis labeled "Time" marked from 9 to 5 in one-hour increments, and the vertical axis labeled "Distance (km)" marked from 0 to 300 in increments of 50. A line connects several points on the graph, starting at $\left(9,0\right)$(9,0), to $\left(11,150\right)$(11,150), to $\left(1,250\right)$(1,250), to $\left(2,250\right)$(2,250), to $\left(3,100\right)$(3,100), and ending at $\left(4,0\right)$(4,0).
When did Ben start his journey?
How far did Ben travel by $11$11 am?
What happened to Ben's speed at $11$11 am?
Ben decreased his speed at $11$11 am.
ABen did not change his speed at $11$11 am.
BBen increased his speed at $11$11 am.
CEvaluate Ben's speed between $11$11 am and $1$1 pm.
What distance did Ben travel between $1$1 pm and $2$2 pm?
What is the furthest distance travelled from the starting point?
What is the total distance travelled by Ben from $9$9 am to $4$4 pm?