Graphs of Motion
Copyright PMDT 2010
Relationship to Junior Certificate Syllabus
Topic
Description of topic
Learning outcomes
Students learn about
Students should be able to
4.5 Relations
Using graphs to represent 
without
phenomena
Formulae
quantitatively.

explore graphs of motion
make sense of quantitative graphs
and draw conclusions from them

make connections between the shape
of a graph and the story of a
phenomenon

describe both quantity and change of
quantity on a graph
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Prior knowledge
Students will have studied patterns and the rules that govern them and will
understand a relation as that which involves a set of inputs,
a set of outputs and a correspondence between each input and each output.
They will be familiar with a multi –representational approach to the
presentation of situations, i.e. the use of blocks, number-strips, tables,
diagrams and graphs.
They will have become familiar with the fact that the X-axis and the Y-axis
are used to represent many other variables as well as these two.
Students will also have studied:
•Graphing of coordinates
•Slope of a line
•The concept of speed as distance/time
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Student Activity 1
Moving from a fixed point at steady speed.
Example 1.1.
The graph below shows Tom’s journey from home to the train station 20km away. Distance
from home is shown on the Y-axis and time in minutes on the X-axis.
Distance from
home(km.)
20
10
1
2
3
0
min
Timeute
(hrs.)
s
(a) How far from home was Tom after 1 hour?
(b) How far from home was Tom after 2 hours?
(c) Was he travelling at a steady speed?
(d) Can you calculate his speed ? Which unit will you use in your answer?
What does speed mean? Discuss this with the other students in your group.
(e) Pick two points on the graph and use them to get the slope of the line.
(f) Compare your answers to (d) and (e). Discuss this with other members of your class.
Conclusion: In a linear time-distance graph the slope of the line gives the speed of the object.
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Example 1.2.
Mary and John run against each other in a 200 metre race. Mary’s time is 20 seconds
and John’s time is 25 seconds. Draw graphs to show their runs using just one set of
scales and axes, assuming that they each ran at a steady speed throughout.
Distance from
start(m.)
Time (secs.)
Calculate the speeds of Mary and John. What units do you use?
Whose graph has a bigger slope ? Calculate the slopes.
How do the graphs show that Mary ran faster than John?
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Student Activity 2
Travel in the opposite direction/staying in the same place
Example2.1
Peter is a runner. He runs from his house at a steady speed until he gets to a marker which
is a distance of 5 km from home, taking 20 minutes to do it. He immediately turns for home
and runs at a slower steady speed, arriving back a further 25 minutes later.
Distance from
house(km.)
Time (min.)
Draw a graph of Peter’s run, showing time on the X-axis and distance from home on the Y-axis.
How will you show his return journey? Discuss this with other members of your group.
Calculate his speed on the outward journey and the return journey. What units did you use?
How do these compare to the slopes of the lines of the graph?
What does the negative slope indicate here?
Examples 2.2, 2.3, 2.4 and 2.5 are other examples of these graphs.
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Example 2.6
Do you remember the story of the hare and the tortoise? Use this graph to re-tell the story:
Distance
from start
T
H
time
Discuss this in your group and write a short story about it.
Student Activity 3
When graphs are not straight lines
Example 3.1
Look at this graph of a horse’s run in a race of 400 metres:
Distance
400
from start
(metres).
300
200
100
10
20
30
40
Time(secs)
How long did the horse take to run the race?
How far did the horse travel in the first 10 seconds?
How far did it travel in the last 10 seconds?
Was the horse travelling at a steady speed throughout?
Describe the speed of the horse from start to finish of the race.
Examples 3.2 and 3.3 are other examples of these graphs.
Example 3.4
Match each of these time –distance graphs with the situation which best fits it.
Situation
Graph
A
Moving at steady moderate pace
1
time
B
Moving at a very fast pace moving
gradually to a slower pace
2
time
3
C
Moving at a fast steady pace
time
D
Moving fast, then slowing slightly,
4
then going faster again.
time
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Student Activity 4
More challenging examples (HL)
Example 4.1
(HL) (Question from PISA)
The following Distance – Time graph shows a comparison between
“walking on the moving walkway” and “walking on the ground next to the moving walkway”.
Person walking on moving walkway
Distance
from start
Person walking on the ground
time
Assume that the walking pace is about the same for both persons.
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Person walking on moving walkway
Distance
from start
Person walking on the ground
time
Consider the following possible speeds and fill in the missing elements in the table:
Put in another possible set of values yourself on the last line.
Which set is most likely to be represented by the graph given?
Now add a line to the graph that would represent the distance versus time for a
person who is standing still on the moving walkway.
Now draw graphs of the other situations in the table and see do you get a
different location for the extra line.
Speed of person on the ground
Speed of walkway
6 km/hr
5km/hr
5km/hr
Speed of person walking on the walkway
11 km/hr
10 km/hr
15 km/hr
Examples 4.2, 4.3, 4.4, 4.5 and 4.6, are other examples of these graphs.
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Time –Velocity (Speed) Graphs
Student Activity 5
Example 1
An express train’s journey from Dublin to Drogheda had the following detail:
In the 1st 10 minutes it built up uniformly from rest to a speed of 60 km per hour
It maintained this speed for 40 minutes
It then reached its destination in the next 10 minutes reducing its speed uniformly to zero in
Drogheda.
Draw a graph of the train’s journey, putting time on the X-axis and speed on the Y-axis.
(this is known as a time-velocity graph or a speed- time graph)
Examples 2, 3, and 4 are other examples of these graphs.
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