 Middle Years

# 12.04 Distance time graphs

Lesson

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.

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?

1. 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. 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.

A

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

B

Ben increased his speed at $11$11 am.

C
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?