Speed, distance and time
Distance and time are two very commonly linked measurements. They
are needed when preparing timetables for trains, planes and buses or for
preparing records for sporting events. From measures of distance and
time you can also calculate velocity and acceleration.
In this section, you will re-examine distance/time graphs to determine
how to gain rich information from them. These graphs are often referred
to a travel graphs.
In most instances data is collected by
measuring distance at regular time intervals.
Therefore distance depends on time. This
means that distance is the dependent
variable and is shown on the vertical axis.
Time is the independent variable and is
shown on the horizontal axis.
Part 2
Distance/time graphs
Distance
As with other graphs, distance/time graphs tell a story. They give much
more information than just a list of data points.
Time
1
There are two common means for measuring distance:
•
distance travelled since time zero (how far you have gone in total)
•
distance from a fixed point (such as home).
It is very important to determine which of these has been recorded on the
graph. For example, the two graphs below both show John’s journey as
he travelled along a straight path. He walked 400 m away from his car in
five minutes, then turned and jogged back to his car in two minutes.
In this first graph, John records how far
he is away from his car at certain times.
The distance increases to 400 m then
decreases more quickly back to zero as
he returns. The second part of the graph
is steeper showing that John was
moving at a faster rate.
In this second graph, John records how
far he has travelled altogether so the
distance is always increasing.
The two different gradients show that he
is travelling more quickly in the second
part of the journey.
d
Distance from car
400
300
200
100
0
d
1
2
3
4
5
6
7 t
Distance travelled
800
700
600
500
400
300
200
100
0
1
2
3
4
5
6
7 t
The first graph can be used to find the total distance by adding the
distance travelled in each section (400 m out and 400 m back). However,
the second graph no longer contains information about the direction John
was travelling, and so is not as useful as the first.
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Part 2 Distance/time graphs
Activity – Speed, distance and time
Try these.
1
The graph below describes how far Lauren is away from home over
a period of five hours. Distance (d) is measured in kilometres and
time (t) is measured in hours.
Distance from home
d
40
30
20
10
0
1
a
2
3
4
5
t
How far is Lauren from her home after one hour and after two
hours? Use this information to write a brief explanation about
what she might have done during this hour?
___________________________________________________
___________________________________________________
___________________________________________________
b
If she was travelling along a straight road, write a brief story to
describe her journey.
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
___________________________________________________
Part 2
Distance/time graphs
3
c
Use the data in the table to find the total distance Lauren
travelled over the five hours.
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
It is important to realise that when recordings of data are made,
information about what has happened between readings is lost. You can
only use the data given. That is why it is vital to take readings as often as
possible when conducting an experiment, particularly when things are
changing in an unpredictable fashion.
Speed
The gradient of a straight line is calculated using the formula
gradient =
rise
run
For distance/time graphs, the gradient of each line segment gives you the
speed because
speed =
distance travelled
time taken
Speed in always considered to be positive, whereas gradients can be
positive or negative. So to interpret the speed from a graph, you can
ignore the sign of the slope.
However, the direction of the slope does tell you the direction of the
journey and is therefore very informative.
Using the graph of Lauren’s journey again, you can calculate various
speeds.
4
Part 2 Distance/time graphs
Distance from home
d
40
30
,
1
so Lauren travelled at an average
30
speed of 30 km/h for the first hour.
The speed for the first hour =
20
10
0
1
2
3
4
5
t
Use the graph to complete the following activity.
Activity – Speed, distance and time
Try these.
2
Calculate Laurens average speed for these parts of her journey.
a
The second hour (between one and two).
___________________________________________________
___________________________________________________
b
The final two hours.
___________________________________________________
___________________________________________________
c
The whole five hours (use total distance travelled).
___________________________________________________
___________________________________________________
Check your response by going to the suggested answers section.
Part 2
Distance/time graphs
5
As with most line graphs, the gradient or slope of the line provides very
useful information. For distance/time graphs, the gradient conveys the
speed and often the direction of travel as well. At a glance, you can see
the fastest or the slowest part of the journey and even time when the
object is at rest (not moving because speed = 0 so the line is horizontal).
The following activity asks you to interpret situations and match them to
possible graphs.
Activity – Speed, distance and time
Try these.
Distance from home
Distance from home
The graphs below are supposed to represent one person’s journey.
One graph shows an impossible situation. State which graph it is,
and describe why it is impossible.
Distance from home
3
Time
Time
Time
Graph A
Graph B
Graph C
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
_______________________________________________________
6
Part 2 Distance/time graphs
4
In each graph below, d represents the distance from home and
t represents time.
d
d
Graph A
t
d
Graph B
t
d
d
Graph D
t
Graph C
t
Graph F
t
d
Graph E
t
Read the descriptions of the journeys below and find the graph that
best matches each journey.
a
Ivan headed off to school from home, but half-way there he
realised he had forgotten his lunch. He ran home, picked up his
lunch and then walked quickly to school.
b
Toby walked up the road from home then ran till he reached the
highway. He turned around and ran towards home, but walked
the last section.
c
John left his uncle’s place and rode his bike straight home.
d
Ali walked to the local café and had morning tea with a friend.
She then caught a taxi to the next suburb and went shopping.
Finally she caught a taxi that dropped her off outside her gate.
e
Amanda ran from her house to her grandmother’s place 2 km
way.
f
Georgie jogged away from home for 2 km, walked for the next
km, turned and headed home. Going home she walked for 1 km
then jogged the rest of the way.
Part 2
Distance/time graphs
7
5
6
Distance from home
Distance from home
Distance from home
Simon drives up the hill at 40 km/h and down the other side at
60 km/h. Which of these graphs best describes this journey?
Time
Time
Time
Graph A
Graph B
Graph C
(Harder) The graph below shows Sue’s journey where d represents
her distance from home.
d
t
Which statement below best describes the journey?
8
•
Her distance is increasing and her speed is increasing.
•
Her distance is decreasing and her speed is increasing.
•
Her distance is increasing and her speed is decreasing.
•
Her distance is decreasing and her speed is decreasing.
•
Her distance is increasing and her speed is constant.
•
Her distance is decreasing and her speed is constant.
Part 2 Distance/time graphs
Check your response by going to the suggested answers section.
The speed is given by the gradient of the line: the steeper the line, the
greater the speed. Also, a positive gradient means the object is heading
away from ‘home’ and a negative gradient shows the object heading
towards ‘home’.
Part 2
Distance/time graphs
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Activity – Speed, distance and time
1
a
After 1 hour she is 30 km from home, and after 2 hours she is
also 30 km from home. There are many explanations as to why
Lauren is in the same place at both times. For example, Lauren
may have stopped for lunch for an hour.
b
Lauren started from home, travelled 30 km in one hour, stopped
for 1 hour, travelled 10 km further away from home in the next
hour then turned around and travelled 30 km back towards home
in the next 2 hours.
c
Total distance travelled = 30 + 10 + 30
= 70 km
2
a
Lauren goes nowhere, so her average speed is zero. The
calculation below shows this using the formula.
distance
time
0
=
1
= 0 km/h
Average speed =
This shows that a horizontal line in a distance/time graph
implies a zero speed. Horizontal lies have a gradient of zero.
b
She travelled 30 km in two hours so her average speed is
15 km/h.
c
Average speed =
total distance
time
70
=
5
= 14 km/h
It is important to note that Lauren may not have actually
travelled at this speed during the journey. This is her average
speed over the five hours.
10
Part 2 Distance/time graphs
Graph C is impossible since at any time the person seems to be in
two places.
Distance from home
3
The person is at both these
places at the same time.
Time
Graph C
4
5
a
Graph C
b
Graph D
c
Graph A
d
Graph F
e
Graph B
f
Graph E
Graph B is correct because the car is always travelling away from
home, and the slope should be steeper for the second section because
the speed is shown by the gradient.
6
Her distance is decreasing and her speed is constant. (The last
statement.)
Her speed is constant because the gradient is the same for whole
length of a single straight line.
Her distance from home is decreasing because the gradient is
negative (she is heading towards home).
Part 2
Distance/time graphs
11
Drawing distance time graphs
In this section you will be asked to create your own distance/time graphs
from descriptions of journeys.
When creating a graph, it is important to remember that the gradient
indicates the speed and direction of the journey.
Activity – Drawing distance/time graphs
Try these.
1
Draw a rough sketch of a distance/time graph to represent each of
these cases.
a
Riding a bike away from home.
b
Riding a bike away from home, stopping because of a flat tyre
and then walking home.
12
Part 2 Distance/time graphs
2
Draw a graph showing a journey where the distance from home is
increasing but the speed is first fast and then slower.
Check your response by going to the suggested answers section.
Graphs can be used to compare two or more sets of data. For the
following activity, you will need to draw more than one line on each
graph so that you can show the difference between two journeys.
Activity – Drawing distance/time graphs
Try these.
Draw two journeys on each rough sketch representing two cyclists both
travelling on the same path. Use the gate at the end of the path as the
zero marker. Label the graph to show which cyclist is represented by
each line.
3
a
Both cyclists travelling at the same speed away from the gate,
with cyclist A starting before cyclist B.
Part 2
Distance/time graphs
13
b
Cyclists A and B travelling in opposite direction at the same
speed then passing one another. Cyclist A starts from the gate.
c
A race from the gate where cyclist A wins.
d
Cyclist B starts from the gate and Cyclist A starts from the other
end of the path at the same time. When they meet, they stop for
a brief chat, then each turns around and goes back to where they
started.
Check your response by going to the suggested answers section.
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Part 2 Distance/time graphs
Creating a picture from a story requires you to understand the meaning of
graphs and, in particular, what the slope and direction of the graph is
telling you. Graphs are a powerful way to convey information and
indicate trends, but are of no use to you unless you know how to read
them.
Activity – Drawing distance/time graphs
1
Your graphs should be similar to these. No scales are required.
a
In this graph, it is important to check that your line goes up from
left to right. The description did not state that the rider actually
started at home so either of these graphs would be correct.
d
d
t
b
t
There are three parts to the journey: fast going away from home,
stopped then a slower section going home. Two possible
answers are shown below.
d
d
t
2
t
‘distance increasing’ means a positive slopes.
‘first fast then slow’ means two different gradients: steep first then
less steep.
Two correct examples are shown below.
Part 2
Distance/time graphs
15
d
d
t
3
t
Your graphs should look similar to these. The important aspects to
check are discussed in each example.
a
‘at same speed’ means the slopes of the two lines should be the
same.
‘away from gate’ means positive slope for both lines.
‘A stating before B’ means that B does not move immediately so
the line B’s line is initially horizontal at d = 0.
d
A
B
t
b
‘opposite directions’ means one line going up and the other
going down.
‘same speed’ means the slopes have the same steepness.
‘passing one another’ means the lines must cross.
Two possible answers are:
d
B
A
t
16
d
B
A
t
Part 2 Distance/time graphs
c
‘from gate’ means starting at the origin (the corner of the axes).
‘A wins’ means the line for A will be steeper because A’s
average speed will be faster.
A
d
B
t
d
‘opposite end of the path’ means that the line for A starts high
up the distance axis.
‘they meet and stop’ means the lines meet and are then
horizontal for a short period.
‘go back to where they started’ means they change direction so
their slopes change direction. B ends up at distance = 0 and A
ends up high up the graph.
The graph below shows one possible answer.
d
A
A
A
B
B
B
t
Part 2
Distance/time graphs
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