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12.07 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?


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.


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?

  1. A




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).

  1. When did Ben start his journey?

  2. How far did Ben travel by $11$11 am?

  3. What happened to Ben's speed at $11$11 am?

    Ben decreased his speed at $11$11 am.


    Ben did not change his speed at $11$11 am.


    Ben increased his speed at $11$11 am.

  4. Evaluate Ben's speed between $11$11 am and $1$1 pm.

  5. What distance did Ben travel between $1$1 pm and $2$2 pm?

  6. What is the furthest distance travelled from the starting point?

  7. What is the total distance travelled by Ben from $9$9 am to $4$4 pm?



interpret distance-versus-time graphs

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