Ch7_Graphs of Motion EDITED

CHAPTER

7 Graphs of Motion

You will learn
istance Time raphs
peed Time raphs

Mohamad Rid uan Pu i, our national Paralympic athlete,
created history by setting a world record in the m
T men s event and the sian ames record in the long ump
event by winning three overall gold medals The best record in
the blue ribbon event, that is s, is a new world record
Rid uan also bro e the long ump sian ames record with
his best personal record of m and won a gold medal in the

metre event ohamad Rid uan Pu i became the first national
athlete to be crowned the est sian Para thlete ale in
a special ceremony held in the nited rab Emirates E on

ebruary
hat is the technique used by runners to win a certain race

Why Study This Chapter?

nowledge about motion is important in the automotive field,
sports science, physics, engineering and astronomy
182

Walking Through Time

WORD BANK Nicholas Oresme

• distance time graph ra ara asa
• speed time graph ra la asa icholas Oresme was an important
• distance ara mathematician and scientist from rance in
• speed la the th century e used the rectangular
• uniform speed la sera a coordinate system and is said to be the first
• deceleration a e ta person who produced speed time graphs
• acceleration e ta
http bt sasbadi com m

183

Chapter 7 Graphs of Motion

7.1 Distance-Time Graphs

What do you understand about distance-time graphs? Learning
Standard
Have you ever used public transport to go to a certain destination?
The ticket for the journey, especially flight ticket, has a display of Draw distance-time
departure time and the estimated time of arrival at your destination. graphs.
For example, the estimated duration of a domestic flight from Kuala
Lumpur International Airport 2 (KLIA2) to Miri is 2 hours 20 minutes.
Do you know how the duration is estimated?

Do you still remember the relationship between speed, distance and
time that you have studied in Form 2?

Speed is a rate which involves distance and time. The relationship between speed and time can
be represented by a distance-time graph. A distance-time graph enables the motion of an object to
be represented in the graphic form which is easy to be understood.

Distance In a distance-time graph:
O
• the vertical axis represents distance

• the horizontal axis represents time

• the gradient of the graph represents the rate of change in
Time distance, that is speed.

CHAPTER 7 How do you draw a distance-time graph?

A distance-time graph can be drawn if the following information regarding the motion is obtained.
(a) Distance-time table
(b) Equation that represents the relationship between distance and time

Draw a distance-time graph based on a distance-time table

1

Azree wants to be a track cyclist and hopes to make Malaysia well-known in the world like
Azizulhasni Awang, our national track cycling champion. The table below shows the distance and
the time taken by Azreen during the training.

Time (minutes) 0 30 60 90 120 Malaysiaku
Distance (km) 0 10 20 30 40
Azizulhasni Awang is
Draw a distance-time graph based on the above table. also nicknamed The
Pocket Rocketman.
Why?

184

Solution: Chapter 7 Graphs of Motion
istance m
Steps
Time
a Choose suitable scales to represent the given distance
and time O minutes

b Plot a point that represents each pair of values of
distance and time on a grid paper or graph paper

c oin the points plotted using a ruler to obtain the
distance time graph as shown

Draw distance-time graphs based on equations that represent the relationship
between distance and time

2

r elva drove his car for a distance of m from uala Lumpur to uantan in hours to visit
his mother The distance, s m, that r elva travelled in t hours is given by the equation s t

raw a distance time graph to represent r elva s ourney from uala Lumpur to uantan

Solution:
Steps

a Construct a distance time table as shown below using the equation s t

Time, t (hours) s t TIPS CHAPTER 7
when t ,
Distance, s (km) s straight line can be drawn
s by oining at least two of
the plotted points
s t
when t , istance m
s
s

b raw the distance time graph as shown by oining
the plotted points



O Time
hours

3

aswinder ingh ta es part in an ilometre cross country event organised by the school during
ational ports ay The relationship between distance from the finishing line and the time of the

run is s t, where s is the distance in m and t is the time in minutes raw a distance time
Saiz sebenar
graph to represent aswinder ingh s run for the duration ഛ t ഛ

185

Chapter 7 Graphs of Motion istance m

Solution: 4
iven s t O

Time, t (minutes) points that satisfy
the equation are
Distance, s (km) enough to draw the
straight line

s t s t Time
when t , when t , minutes
s s
s s

CHAPTER 7 7.1a

1. The table below shows the time ta en by a i li to wal from his house to the mosque for
prayers e needs minutes to wal to the mosque which is located m from his house
raw a distance time graph based on the given table

Time (minutes)

Distance (metres)

2. Enci yambe drives his car to his office which is located m from his house in e enu
The table below shows the time ta en by Enci yambe to reach his office in iri from
e enu raw a distance time graph based on the given table

Time (hours)

Distance (km)

3. The motion of a particle for a certain period is represented by s t where s is the distance
in cm and t is the time in seconds raw a distance time graph to represent the motion of the
particle for a period of seconds

4. Leong cycles to ainal s house which is located m from his house Leong s ourney from
ainal s house is given by the equation s t where s is the distance in m and t is

the time in minutes raw a distance time graph to represent Leong s ourney for the period
ഛ t ഛ

186

Chapter 7 Graphs of Motion

How do you interpret distance-time graphs? Learning
Standard
tudy the two distance time graphs below
Interpret distance-time
istance metres istance metres graphs and describe the
motion based on the
SC P graphs.

s

Time R Time MY MEMORY
seconds T seconds
O t t T O t t Speed = Distance
Time
iagram iagram
Gradient

iagram shows a motion iagram shows a motion = Vertical distance
from O to C for a distance from O to R passing through Horizontal distance
of S metres in a period of P and

seconds

OA positive gradient positive gradient TIPS

of graph of graph Positive gradient and
negative gradient of
gradient of OA motion for a distance a distance time graph
indicate directions
represents speed of of S metres in a period of motion

motion of t seconds Smart Mind

motion at uniform The distance-time
graph below shows the
speed motion of an object.

motion for a distance istance metres

of s metres in a period zero gradient s
of t seconds no change in distance Time

stationary O t seconds CHAPTER 7
What type of equation
zero gradient stationary for the period will give us the
no change in distance t t seconds distance-time graph
as shown above?
means motion stops Discuss and state one
example of the motion.
stationary negative gradient
negative speed shows
stationary for the period
t t seconds
ob ect moves bac to

original place or

moves in the opposite

positive gradient direction

motion continues to C

motion of S metres in OR motion of S metres
a period of T seconds to and fro in a period
of T seconds

187

Chapter 7 Graphs of Motion

4 istance m A
Car
The distance time graph shows the motion of a car and a tour Tour bus
bus raph OA represents the motion of the car from Puchong O
to ela a raph represents the motion of the tour bus C Time
from ela a to Puchong etermine the difference in speed, hours
in m h , of the two vehicles

Solution:

peed of car peed of tour bus cdoirfrfeesrepnocned iinng d disitfafenrceen tcrea vine ltliemd e

cdoifrfreersepnocned iinn gd idsitfafnecreen tcraev ienl lteimd e
ho umrs
m The negative sign
hours
means the tour bus m per hour
and the car move
m per hour in the opposite
direction m h
m h

ence, difference in speed m h TIPS
m h
m h can also be written
5 as m h

The distance time graph shows the motion of a car for istance m
a period of hours
a etermine
CHAPTER 7 i the duration when the car is stationary
ii the speed, in m h , of the car in the first hour
b escribe the motion of the car for the last minutes

Solution: Time
hours
a i tationary period period the car stops O

hour

hour rom the graph, the
distance travelled in the
ii peed of car in the first hour speed of car in the first hours first hour is not nown
Therefore, the speed is
m determined by the distance
hours travelled in the first
hours The gradients
m h on a straight line are
peed of car m the same

b h

}

m h last minutes

Saiz se benTaher car travels for m with a speed of m h in the last minutes

188

Chapter 7 Graphs of Motion

6 Distance (km)
2
Sahana cycles to the post office to send Hari Raya Aidilfitri
greeting cards to her close friends. The distance-time graph O t Time
shows Sahana’s journey from her house to the post office 0920 0950 1010 (24-hour
and back. system)

(a) Determine

(i) the total distance, in km, for Sahana’s whole journey.
(ii) the value of t, if Sahana cycles with a speed of

8 km h–1 to the post office.

(b) Describe Sahana’s journey from the post office back to
her house.

Solution: (ii) Time = Distance
Speed
(a) (i) Total distance travelled = 2 km + 2 km
= 4 km 2 km
8 km h–1
(0 – 2) km =
1010 – 0950 hour
(b) Rate of change in distance =
( )1 = 0.25 hour
minute = 1 hour 60 = 15 minutes
60 = – 6 km h–1
Thus, t = 0920 + 0015
= 6 km h–1 = 0935

Sahana cycles for a distance of 2 km in 20 minutes with TIPS
a speed of 6 km h–1.
Speed = —DTi—sitma—nec—e
Time = —DSi—sptea—endc—e

Motion with different speeds CHAPTER 7

The speed of a motion usually changes in a journey. In such situation, average speed is used.

Average speed = Total distance
Total time

7 Distance (km)

Zabedah wants to visit her friend in Muar. The distance-time 80 Muar
graph shows her journey by car from Segamat to Muar passing Tangkak
through Tangkak. Time
s (hours)
(a) Calculate the average speed, in km h–1, of Zabedah’s 25
journey from Segamat to Muar.
O 0.5 1.0 1.5
(b) If the rate of change in distance of the car from Segamat Segamat
to Tangkak is 50 km h–1, calculate the distance, in km,
between Tangkak and Muar.

(c) Describe the motion of the car from Segamat to Muar.

189

Chapter 7 Graphs of Motion

Solution: TIPS

a verage spe ed ToTtoa lt adli tsitmanece peed istance
Time
m
h istance peed × Time

m h

b Total distance verage speed × Total time c The car moves for a distance
of m in hours with
m h × h
an average speed of m h
m istance between

egamat and

Tang a



istance between Tang a and uar m

m

7.1b

1. The distance time graph shows the ourney of Enci istance m
Re ab and his family by car from ota inabalu to
eningau to celebrate esta aa ata

a Calculate the speed, in m h , of the car for the
last hour

b escribe the motion of Enci Re ab s car for Time
the period of minutes after travelling the first hours
m of the ourney
Time
CHAPTER 7 c i Calculate the average speed, in m h , O minutes
for the ourney from ota inabalu
to eningau

ii ence, describe the motion of the car for the
whole ourney

2. Enci Rashid exercises every day to stay healthy istance m
The distance time graph shows the distance and O
time of Enci Rashid s run from his house to the
playground and then bac to his house

a Calculate the difference between the speed
of Enci Rashid s run from his house to the
playground and the speed of his run from the
playground bac to his house in m h

b Calculate the average speed, in m h ,
for Enci Rashid s whole run

c escribe Enci Rashid s motion for the period
of minutes

190

3. The distance time graph shows Puan Ro ita s ourney Chapter 7 Graphs of Motion

by car for a period of 4 hours represents Puan istance m
Ro ita s ourney from her wor place to a mar et and R

RS represents the return ourney to her house P S Time
O hour
a Calculate the value of t, if the speed of the car for
Puan Ro ita s ourney from her wor place to the t system
mar et is m h
istance m
b escribe the motion of the car represented by
i the straight line . A
ii the straight line .
Time
4. Enci usri wor s in a law firm Every day Enci O t minutes
usri sends his child to school on his way to wor place

by car OA represents the ourney from his house to
school and represents the ourney from the school
to his wor place

a Calculate the value of t, if the rate of change in
distance of the car from the school to his wor place
is m h

b escribe the motion of the car for the whole ourney
from his house to his wor place

How do you solve problems involving distance-time graphs?

8 Learning CHAPTER 7
Standard
The incomplete distance time graph shows r Tan s ourney from
eremban to Lumut r Tan stops at Rawang for lunch and a short Solve problems involving
distance-time graphs.
brea before he continues his ourney to Lumut

a f the average speed of r Tan s car from eremban to

Rawang is m h , calculate the distance, in m, between ist ance Rawang
eremban and Rawang m

b The distance between eremban and Lumut is m and R Time
r Tan drives at an average speed of m h to reach Lumut hours
O
from Rawang Complete the given distance time graph to

represent r Tan s whole ourney



eremban

191

Chapter 7 Graphs of Motion

Solution:

Understanding problem Planning strategy

a Calculate the distance between a peed istance
eremban and Rawang in m istance Time
peed × Time
b Complete the distance time graph
from Rawang to Lumut b • etermine the distance between Rawang and

Lumut

• C T iommep le te Titshitmea ndecisetance time graph
•

Carrying out strategy istance m
O
a istance peed × Time

m h × hours

m

b istance between Rawang and Lumut

m m m
istance
Time peed

m Time
m h hours

hours

CHAPTER 7 Conclusion hours

a The distance between eremban and Rawang is m
b The distance between Rawang and Lumut is m and the time ta en is

9

The distance time graph shows the ourney of two cars istance m
P
between uala Lipis and Cameron ighlands raph C
A Time
represents Enci anaf s ourney together with his family
t minutes
from Cameron ighlands to uala Lipis to attend his cousin s O

wedding raph represents the ourney of Enci Raven s

family from uala Lipis to Cameron ighlands for a holiday

a The rates of change in distance for OA and are the
same Calculate the value of t

b The average speed of Enci anaf s ourney is m h

Saiz sebenCaalrculate the difference in time, in minutes, for the two

ourneys to reach their respective destinations

192

Chapter 7 Graphs of Motion

Solution:

Understanding problem Planning strategy

a Calculate t, that is time in minutes a radient of OA radient of

b ifference in time of the two cars b • Time ta en by Enci Raven t
ourney to reach their respective
destinations • etermine the time ta en by

Enci anaf

• Time istance
peed

Carrying out strategy

a m t m b • Total time for Enci Raven s ourney,
min m in t minutes

t • Total time for Enci anaf s ourney in minutes

T ime m
t m h

hours ×
t
minutes

t • ifference in time

minutes

Conclusion CHAPTER 7

a t
b The difference in time for the two ourneys to reach their respective destinations

is minutes

7.1c

1. The distance time graph shows the time ta en by the istance metres

two best participants in the metre event during EC
A
the ports Championship of inar arapan Time
O seconds
raph OE represents Ri al s run and graph
193
represents effery s run represents the time

elapsed before effery continued his run after a fall

a Calculate the time lost, in seconds, by effery
in the competition

b id effry have the chance to become the
champion in the metre event if he did not
fall down and manage to maintain his speed for
the whole run ustify your answer

Chapter 7 Graphs of Motion

2. The incomplete distance time graph shows Enci umali s istance m Time
ourney for a distance of m A minutes
O t t
a t is given that the rate of change in distance for the
first m is m h Calculate the value of t

b f Enci umali s car is stationary for minutes,
calculate the value of t

c is ourney continues from to its destination
with an average speed of m h Complete the
distance time graph for the whole ourney of Enci
umali

d f the ourney from O starts at in the morning,
calculate the time Enci umali arrives at his
destination

3. Enci amal goes to Padang esar with his family uring istance m Time
the return ourney to itra, they stop at u it ayu itam P t minutes
for a tea brea The distance time graph shows the return O
ourney from Padang esar to itra
CHAPTER 7
a Calculate the duration in which Enci amal s car
is stationary

b t is given that the average speed for the ourney
from Padang esar to u it ayu itam is
m h

i etermine the value of

ii Calculate the distance between Padang esar and
u it ayu itam

c f Enci amal drives at an average speed of
m h for the return ourney from u it ayu

itam to his house in itra, calculate the value of t

d Calculate the average speed, in m h , for the
whole ourney

4. The distance time graph shows r oorthy s ourney istance m
by car for a distance of m in t minutes t is given
that the rate of change in distance before and after the
stationary period are the same

a Calculate the value of t

b Calculate the average speed, in m h , for the whole
ourney of r oorthy

c escribe the motion of the car after the
stationary period

Time
O t t t minutes

194

Chapter 7 Graphs of Motion

7.2 Speed-Time Graphs

What do you understand about speed-time graphs? Learning
Standard
Have you observed the motion of the needle in a speedometer of
a car when your parents drive the car? The needle shows that the value Draw speed-time graphs.
of the speed changes when the oil pedal or brake pedal is pressed.
Study the diagrams of the speedometer below.

Diagram 1 Diagram 2 Diagram 3 Diagram 4

The needle in Diagram 1 shows a value of 0, which means the vehicle is stationary. The increase in
value as shown in Diagram 2 and Diagram 3 means the speed of the vehicle is increasing. The speed
in Diagram 4 is decreasing compared to the speed shown in Diagram 3 for a certain period. The rate
of change of speed of a motion can be shown by drawing a speed-time graph.

Speed In a speed-time graph:
O • the vertical axis represents the speed of a motion.
• the horizontal axis represents the time taken.
• the gradient of graph represents the rate of change of speed,
Time that is acceleration.

How do you draw a speed-time graph? CHAPTER 7

A speed-time graph can be drawn if the following information about the motion is obtained.

(a) Speed-time table
(b) Equation that represents the relationship between speed and time

How do you draw a speed-time graph based on a speed-time table?

10
The table below shows the change of speed of Encik Azizul’s car for a period of 5 seconds.

Time (seconds) 0 1 2 3 4 5
Speed (m s–1) 0 10 20 30 40 50
Draw a speed-time graph based on the given table.

195

Chapter 7 Graphs of Motion

Solution: Speed (m s–1)

Steps

(a) Choose appropriate scales to represent the 40
given speed and time. 30
20
(b) Plot a point to represent each pair of values of
speed and time on a grid paper or graph paper.

(c) Join the plotted points using a ruler to obtain 10
the speed-time graph as shown.

O Time
(seconds)

11

The rate of change of speed of an aeroplane that is landing is given by the equation v = 800 – 1 600t
where v is the speed in km h–1 and t is time in hours. Draw a speed-time graph to represent the
landing of the aeroplane for the period 0 ഛ t ഛ

Solution:
Steps

(a) Construct a speed-time table as shown below using the equation v = 800 – 1 600t.

Time, t (hours) 0 0 Speed (km h–1)
Speed, v (km h–1) 800 800

CHAPTER 7 v = 800 – 1 600t v = 800 – 1 600t O
when t = 0, when t ,
v = 800 – 1600(0) v
v = 800 v=0

(b) Draw the speed-time graph by plotting the points based on Time
the table constructed. (hours)

7.2a

1. Draw a speed-time graph based on the given table.

(a) Time (seconds) 0 1 2 3 4 (b) Time (minutes) 0 1 2 3 4
Speed (km min–1) 30 20 10
Speed (m s–1) 3 4 678

2. Draw a speed-time graph by constructing a speed-time table for each of the following equations.

It is given that v is the speed in m s–1 and t is the time in seconds.

(a) v = 60 – 2t ; 0 ഛ t ഛ 30. (b) v = 3t ; 0 ഛ t ഛ

196

Chapter 7 Graphs of Motion

What is the relationship between the area under Learning
a speed-time graph and the distance travelled? Standard

The diagrams below show two graphs: Make a relationship
between the area under
Distance Speed speed-time graph and
the distance travelled,
sv and hence determine
speed (v) the distance.

O t1 Time O t1 Time
Diagram 1 Diagram 2

From the gradient of the distance-time graph in Diagram 1, we can determine the speed of the
motion. This information from the distance-time graph can be used to draw the speed-time graph
as shown in Diagram 2. Do you know that the distance, s, travelled by a motion can be determined
from the speed-time graph?

Mind Stimulation 1

Aim: To determine the relationship between the area under speed-time graph and the distance
travelled.

Steps:

1. Divide the class into groups.
2. Read and understand each of the given statements. Calculate the average speed in km h–1.
3. Sketch the speed-time graphs based on the given statements.
4. Calculate the area under the speed time graph and the distance travelled li e example a

Statement Speed-time graph Area under the graph Distance travelled

a Enci ai al drives Speed (km h–1) Area Distance travelled

a distance of 200 km = 4 h × m h–1 = speed × time CHAPTER 7
in 4 hours. = 200 km m h–1 × 4 h

Speed = 200 km Time = 200 km
4 hours 4 (hours)
O

m h–1

(b) A tour bus moves
m in hours

(c) Mrs Malini cycles
8 km in 40 minutes.

d r ome runs
4 km in 30 minutes.

Discussion:

1. What is the relationship between the area under speed-time graph and the distance
travelled?

2. Present your group s findings through allery al activity
3. re your group s findings the same as the other groups findings

197

Chapter 7 Graphs of Motion

From the activity in Mind Stimulation 1, it is found that:

The area under a speed-time graph is the same as the distance travelled for
the same time interval.

In general,

For a speed-time graph:
Area under the graph = Distance travelled

12

Calculate the distance travelled in each motion based on the following speed-time graphs.

(a) Speed (km h–1) (b) Speed (km h–1)

10 Time 80 Time Indicator
—4 (hours) 40 6 (minutes)
O Make sure the unit
O for time used in the
Solution: speed is the same
as the unit for time.

(a) (b)

CHAPTER 7 10 80
4 40

(6 ÷ 60)

istance area of trape ium istance area of trape ium

1( )= × 4 h × m h–1 1( )=× 6 h × m h–1
2 2 60

= 6 km = 6 km

13 Speed (km h–1)
72
The speed time graph shows the speed of rs Liew s car for
a period of 36 minutes. Calculate 40

a the total distance, in m, travelled by rs Liew s car
for the period of 36 minutes.

(b) the average speed, in km h–1, of rs Liew s car for the O 10 20 36 Time
period of 36 minutes. (minutes)

198

Chapter 7 Graphs of Motion

Solution:

(a) Total distance

= area under the graph
[ ] [ ] [ ]( ) ( ) ( )=
1 × 10 h × 72 km h–1 1 × 10 h × m h–1 1 × 16 h × m h–1
2 60 2 60 2 60

( )= 238 224 km

= 30.27 km

(b) Average speed = Total distance MY MEMORY
Total time

= 30.27 km 60 minutes = 1 hour
(36 ÷ 60) h
m h–1 1 minute = 1 hour
60

7.2b

1. Calculate the distance travelled, in km, in each motion based on the given speed-time graph.

(a) Speed (km h–1) (b) Speed (m s–1) (c) Speed (km h–1)

20 16
60

10 8

O2 Time O Time O Time
6 8 (hours) (seconds) (minutes)

Speed (km h–1) Time CHAPTER 7
30 (minutes)
2. The speed time graph shows the speed of Enci ustaffa s 40
motorcycle for a period of 30 minutes when he fetches his

child after extra class Calculate

(a) the total distance, in km, for a period of 30 minutes. 20
(b) the average speed, in km h–1, for a period of

30 minutes.

O8 18
Speed (m s–1)
3. Sarves competes in the 100-metre event during the sports
event in his school. The speed-time graph shows the speed 12
of Sarves up to the finishing line. Calculate 8
(a) the value of t.
(b) the average speed of Sarves in km h–1.

Time
O t – 6 t (seconds)

199

Chapter 7 Graphs of Motion

How do you interpret speed-time graphs? Learning
Standard
Study the speed-time graph below.
Interpret speed-time
Speed (m s–1) Uniform speed graphs and describe the
movement based on
Acceleration R the graphs.
vQ
INFO ZONE
P Deceleration
u Distance
• The length of the space
A BC
between two points
S Time (s) Displacement
O t1 t2 t3 • Vector distance from

The interpretation of the speed-time graph: a fixed point measured
in a certain direction
PQ The speed of object increases from u m s–1 to v m s–1.

The gradient of graph is positive, hence the rate of
change of speed is positive.

Acceleration = Change of speed INFO ZONE
Change in time

The area of trape ium A, that is the area under graph Speed
PQ represents the distance travelled in the period of • Rate of change in
t1 seconds.
distance with respect
to time.

Distance
• Speed = Time

QR There is no change of speed ero gradient Velocity

The object moves at a uniform speed. • Rate of change in

The area of rectangle B, that is the area under graph displacement with
QR represents the distance travelled in the period of
(t2 – t1) seconds. respect to time.

CHAPTER 7 • Velocity = Displacement
Time

RS Speed of the object decreases. MY MEMORY

The gradient of graph is negative, hence the rate of Speed = Distance
change of speed is negative. Time

Change of speed Acceleration = Speed
Change in time Time
Deceleration =

There is no change in the direction, that is the motion of Indicator
the object remains in the same direction.
The direction of
The area of triangle C, that is the area under graph motion of an object
RS represents the distance travelled in the period of remains the same
(t3 – t2) seconds. during acceleration
or deceleration.

200

Chapter 7 Graphs of Motion

14 Speed (m s–1) Time
60 (seconds)
The speed time graph shows the motion of Puan alina s car 40
for a period of seconds 20

(a) Calculate the rate of change of speed, in m s–2, for the O
first seconds

b escribe the motion of the car for the second seconds
(c) Calculate the total distance, in metres, travelled in the

period of seconds

Solution:

Change of speed (b) The car moves at a uniform
(a) Rate of change of speed = Change in time speed of 40 m s–1 for the

= (40 20) m s–1 period of seconds
0) s

= 4 m s–2

(c) Total distance travelled = area under the graph

1΄ ΅= ΄ ΅ 1
2 2

m

= 600 m

15

Mr Daniel Wong drives his car to a convenience store to buy newspaper. The speed-time graph
shows the motion of the car from his house to the road junction before reaching his destination.

a escribe the motion of r aniel ong s car for the first Speed (m s–1)
10 seconds.
CHAPTER 7
b hat happens to the motion of r aniel ong s car
from the 10th second till the 20th second?

(c) Calculate the rate of change of speed, in m s–2, for the last
seconds

(d) Calculate the distance, in metres, travelled during O Time
deceleration and describe the motion of the car for (seconds)
the period.

Solution:

(a) Rate of change of speed for the first 10 seconds

= Change of speed
Change in time
0) m s–1
= 10 0 s

m s–2 Saiz sebenar

The car accelerates at a rate of m s–2 for the first 10 seconds. 201

Chapter 7 Graphs of Motion

b r aniel ong s car moves at a uniform speed of m s–1 from the 10th second till the
20th second.
The rates of change in speed for the last
(c) Rate of change of speed = (0 m s–1 seconds and for the last seconds
20) s are the same.

= – 3 m s–2

d istance travelled during deceleration distance travelled in the last seconds

= 12 The answer can be

written as 3 m s–2

΄ ΅=1 m • acceleration
2 or
• deceleration 3 m s–2
m

The car travels for m in seconds with a deceleration of 3 m s–2.

7.2c Speed (m s–1)

1. The speed-time graph shows the motion of a motorcycle 20
for a period of seconds escribe the motion of the
motorcycle O Time
(a) for the first 20 seconds. Speed (m s–1) (seconds)
(b) when it moves at a uniform speed. 20 Time
(seconds)
CHAPTER 7 2. The speed-time graph shows the motion of a particle for 10 Time
a period of 18 seconds. (seconds)
(a) Calculate the acceleration, in m s–2, of the particle for O
the last 6 seconds.
(b) Calculate the total distance, in metres, travelled by the Speed (m s–1)
particle in the period of 18 seconds.
(c) Describe the motion of the particle when it moves at O
a uniform speed.

3. Encik Merisat visits his friend who lives in the same
housing estate by car. The speed-time graph shows the
ourney of Enci erisat to his friend s house
(a) Calculate the rate of change of speed , in m s–2, of
his car for the first 20 seconds.
(b) Calculate the distance, in metres, travelled at
a uniform speed.

c escribe Enci erisat s ourney for the period of
minutes

202

Chapter 7 Graphs of Motion

How do you solve problems involving speed-time graphs?

16 Learning
Standard
The speed-time graph shows the motion Speed (m s–1) t
of a car for a period of 14 seconds. Calculate v Solve problems involving
speed-time graphs.
(a) the average speed, in m s–1, for the first 10
6 seconds. Time
O 46 14 (seconds)
(b) the value of t, if the distance travelled by
the car for the first 4 seconds is half the
distance travelled at a uniform speed.

(c) the value of v, if the acceleration for the
last seconds is m s–2.

Solution:

Understanding the problem Planning a strategy

(a) Average speed for the first (a) Average speed = Total distance
6 seconds. Total time

(b) Value of t, that is the time when (b) Distance for the first 4 seconds
moving at a uniform speed. 1
= 2 (distance travelled at a uniform speed)
(c) Value of v, that is the final speed
when the acceleration is m s–2. (c) Acceleration = Change of speed
Change in time

Carrying out strategy

(a) Total distance travelled for the first b rea of trape ium 1 (area of rectangle)
2
6 seconds
1 4 21 × (t – 4) × 10 CHAPTER 7
΄ ΅=1 × 10] 2 × ×
2 × 4 ×
t – 20
m
t
m
t = 10

Average speed = —6 —sm c cceleration m s–2

v 1100
14

= —3 m s–1 v 410

v – 10 = 14
v = 24

Conclusion

(a) The average speed for the first 6 seconds is 3 m s–1.
(b) t = 10

(c) v = 24

203

Chapter 7 Graphs of Motion

17 Speed (m s–1)

The speed-time graph shows the1m7 otion of two vehicles. Graph v A D
OAB represents the motion of Enci abadi s car and graph v – 10 B
CD represents the motion of a taxi driven by r Low The
difference between the distance travelled by the car and the taxi C Time
in the period of 24 seconds is 160 m. Calculate the value of v. O8 24 (seconds)

Solution:

Understanding the problem Planning a strategy

• alue of v, that is the final speed of the taxi • istance travelled by the car distance
in the period of 16 seconds. travelled by the taxi m

Carrying out strategy

Distance travelled by the car (OAB distance travelled by the taxi CD) = 160

΄ ΅ ΄ ΅1×(v– 10) × – 1 × (24 – 8) × (v) = 160
2
2

΄ ΅ ΄ ΅1×(v– 10) × 40 – 1 × 16 × v = 160
2
2

20v – 200 – 8v = 160

12v = 360

v = 30

Checking Answer

Conclusion • istance travelled by the car 1 × (30 – 10) × (24 + 16) = 400 m
Value of v = 30 2
1
CHAPTER 7 • istance travelled by the taxi 2 × 16 × 30 = 240 m

• ifference in distance – 240 = 160 m

18 Speed (m s–1)
30
The speed-time graph shows the motion of a van for a period
of t seconds. Calculate 12
(a) the rate of change of speed, in m s–2, for the first 3 seconds. O3
(b) the distance, in metres, travelled for the first 10 seconds.
(c) the value of t, if the magnitude of the rate of change of

speed after 10 seconds is the same as the magnitude of the
rate of change of speed for the first 3 seconds.

Solution: Time
t (seconds)
(a) Rate of change of speed = (30 – 12) m s–1 10
(3 – 0) s

Saiz sebenar = 6 m s–2

204

΄(b) Distance travelled = 1 × 3 × ΅ × 30] Chapter 7 Graphs of Motion
2
Observe the
m speed limits for
the safety of all.
= 273 m

(c) Acceleration after 10 seconds = acceleration for the first 3 seconds

(0 – 30) m s–1 (30 – 12) m s–1
(t – 10) s (3 – 0) s
΄ ΅– =

( )–
–30 = 18
t – 10 3

The rate of change of t 30 = 6 INFO ZONE
speed is in the same – 10
direction (magnitude). Magnitude
30 • distance travelled in
6 = t – 10
a certain direction.
t

7.2d Speed (m s–1)
v
1. The speed time graph shows the motion of ion ohan s
car for a period of 16 seconds. Calculate 12
(a) the distance travelled, in metres, at a uniform speed.
(b) the value of v, if the average speed of the car for the O Time
first 12 seconds is 14 m s–1. 8 12 16 (seconds)

2. The speed-time graph shows the motion of the motorcycle Speed (m s–1) CHAPTER 7
ridden by Abit Lusang for a period of t seconds. Calculate 12
(a) the deceleration of the motion in m s–2. 8 Time
(b) the distance, in metres, when the rate of change of t (seconds)
speed is positive. O3 7
(c) the value of t, if the total distance in t seconds Speed (m s–1) Time
is m t (seconds)
10
3. The speed-time graph shows the motion of a car for 6
a period of t seconds. Calculate 2
(a) the total distance, in metres, when the rate of change of O 3 8 10
speed is positive.
(b) the value of t, if the magnitude of the rate of change of
speed from the 8th second till the 10th second is the
same as the magnitude of the rate of change of speed
after 10 seconds.

205

Chapter 7 Graphs of Motion

1. The distance-time graph shows the motion of an express Distance (km)
bus for a period of 14 minutes. Calculate 10
(a) the duration when the bus is stationary. 6
(b) the rate of change in distance, in km h–1, of the bus
for the last 4 minutes. O Time
(c) the average speed, in km h–1, of the bus for the 4 10 14 (minutes)
period of 14 minutes.

2. A car and a tour bus travel for 150 km in 3 hours. The Distance (km)
distance-time graph shows the motion of the car and
the tour bus. Calculate 150 A Tour bus R
(a) the rate of change in distance, in km h–1, of the car Car
for the first 42 minutes. 70 P Q
(b) the value of t.
(c) the rate of change in distance, in km h–1, of the car O 0.7 t B Time
for the last 80 km. 3 (hours)

3. The speed-time graph shows the motion of a car and Speed (m s–1)
a motorcycle. Calculate
(a) the duration when the motorcycle travels at 10 R
a uniform speed. Car
(b) the value of t, if the distances travelled by the car
and the motorcycle are the same for the period of 4P Q Motorcycle
t seconds.
O2 Time
10 t (seconds)

CHAPTER 7 4. The speed-time graph shows the motion of a particle for Speed (m s–1)
a period of 15 seconds. Calculate 8
(a) the rate of change of speed, in m s–2, of the particle 7 Time
for the last 6 seconds. v 15 (seconds)
(b) the value of v, if the ratio of the distance travelled
in the first 5 seconds to the last 6 seconds is 5 : 3. O 59
(c) the average speed, in km h–1, of the particle for the
period of 15 seconds.

5. The speed-time graph shows the motion of two cars for Speed (m s–1)
a period of 45 minutes. Puan Nisha drives from Town P
to Town Q and Puan Farah drives in the opposite 24 Town Q
direction. Calculate 18
(a) the value of v, if the rate of change of speed of
Puan Farah’s car for the first 30 minutes is the v
same as the acceleration of Puan Nisha’s car
for the period of 45 minutes. O 30 Time
Town P 45 (minutes)

206

Chapter 7 Graphs of Motion

(b) the distance from Town Q, in km, when the two vehicles meet.
c the time, in minutes, ta en by Puan isha to reach Town Q if the acceleration of her car

does not change.

6. The distance-time graph shows the motion of two cars Distance (km) A B
for a distance of 100 km. Graph OA represents the 100
motion of the car driven by Mr Lee at an average speed of
v km h–1 and graph OB represents the motion of the O Time
car driven by Encik Dollah at an average speed of 0800 (24-hour
(v – 20) km h–1. Calculate t min t system)
(a) the value of v, if the difference between the time
taken by Mr Lee and Encik Dollah to reach the Distance (km)
destination is minutes 1
(b) the time, in the 24-hour system, Mr Lee reaches
his destination. O Diagram 1 t Time
Speed (m s–1) (seconds)
7. (a) (i) Diagram 1 shows the distance-time graph of
car A for a period of t seconds. It is given that v
the average speed of car A is m s–1. Calculate
the value of t. O Diagram 2 t Time CHAPTER 7
(seconds)
(ii) Describe the motion of car A for the period of
t seconds.

(b) Diagram 2 shows the speed-time graph of car B.
It is given that the uniform speed, v, of car B is
the same as the average speed of car A and both
vehicles travel the same distance. If the values of t,
in seconds, for both graphs are the same, calculate
the duration, in minutes, during which car B moves
at a uniform speed.

PROJ ECT

1. Collect information about the various speed limits in the area where you stay.
2. Record the speed limits according to specific areas or example, m h–1 at the

surrounding areas of a school.
3. What is the implication if a driver does not follow the speed limit?
4. Prepare a report with photos and present your findings using multimedia.

207

Chapter 7 Graphs of Motion

CONCEPT MAP
Graphs of Motion

Distance-Time Graphs Speed-Time Graphs

Distance B Speed Q
sA VP

O C Time O t1 t2 R Time
t3 t3
t1 t2
• Distance travelled = Area under graph
Change in distance
• Gradient = Change in time • Gradient = Change of speed
Change in time

Rate of change in distance = speed Rate of change of speed = acceleration
• OA positive gradient
• OP positive gradient
(motion towards destination)
• AB zero gradient speed increases

(stationary) acceleration
• BC negative gradient
• PQ zero gradient
(motion towards origin)
• Positive and negative speeds indicate no change of speed

opposite direction of motion. uniform speed

• QR negative gradient

speed decreases

deceleration

CHAPTER 7 Distance (km) B Speed (m s–1) B
50 A 30 A

O C Time (hours) O C Time (seconds)
123 40 100 150

(a) Speed of the object in the first hour or (a) Acceleration in the first 40 seconds

in the last hour = 30 − 0 = 0.75 m s–2
40 0
50 − 0
= 1−0 = 50 km h–1 (b) Acceleration in the last 50 seconds

(b) The object is stationary for 1 hour (AB) = 0 − 30 = – 0.6 m s–2
150 − 100

(c) Uniform speed for 1 minute (AB)

Total distance
Average speed = Total time

208

Chapter 7 Graphs of Motion

Self Reflection

Instructions
1. ill in the boxes with the correct answers using capital letters
2. ill in the mystery word pu le according to the numbered boxes
3. ow do you feel with the completed mystery word pu le

(a) Change in distance =
Change in time

b or a distance time graph, ero gradient means the state

of 4 1

(c) ————Ch—an—ge–—of—sp—eed———— = Acceleration

Change in 6

(d) For a speed-time graph, negative gradient means 5

3

(e) A uniform speed of a motion occurs when the value of the
2 is ero in a speed time graph

(f) Average 7 = Total distance CHAPTER 7
Total time

Mystery Word Puzzle

4 26 623217425

1. Divide the class into groups.
2. Each group is required to collect information about high-speed vehicles that travel on

the surface of the earth, in the air and outer space from various sources.
3. The information obtained should include the history of the invention, applications as

well as side effects.
4. Exhibit your group s findings at the athematics Corner for the benefits of others

209

(c) y (c) minimum = 10, maximum = 30

y = –x y=x–4 (d) RM2 625 1
y=2 2 x 2
8. (a) x + y ഛ 1 000, y ജ x
–2 O 24
–2 (b) y

1 000 (d)(ii)

–4

3. (a) y ജ –2x, y Ͼ x and y Ͻ 4 750
1 500 y = —12 x
(b) y Ͻ 2x, y ജ 2 x and y ഛ – 1 x + 6
2
(c) y – x ഛ 4, 2y Ͼ x + 4 and y Ͻ 3

(d) y ജ 3 x + 6, x Ͼ – 4, y Ͻ 5 250 x + y = 1 000
2

4. (a) y O x

y=x 250 500 750 1 000

10 y = 10

y + x = 10 (c) minimum = 250 m, maximum = 500 m
10
(d) (i) y ജ 3x
(ii) Refer to the graph.

O x CHAPTER 7 Graphs of Motion
(b)
Self Practice 7.1a
y = –x
y 1. Distance (metres) 2. Distance (km)
y=x+6
300 45
6 y = —32 x + 4 225
4 36

27

150 18

75 9

Time Time
O 5 10 15 20 (minutes) O 0.2 0.4 0.6 0.8 1.0 (hours)

–6 O x 3. Time, t (seconds) 0 5 4. Time, t (minutes) 0 8

5. (a) y ഛ 2, x Ͻ 3, y ജ – x, y ജ 0 Distance, s (cm) 5 45 Distance, s (km) 1.6 0

(b) y Ͼ –2x, y ജ 2x – 8, y ഛ – 12x Distance (cm) Distance (km)
50
6. (a) y Ͻ –1, 1.6
(b) x ജ 2, 40
x ജ – 5, y ജ 4 x – 1 1.2
5 30
0.8
y ജ 0, y Ͻ –x + 6 20
0.4
10
7. (a) x + y ഛ 40, y ജ 10, x ഛ 25 Time Time

(b) y (moden) O 1 2 3 4 5 (seconds) O 24 6 8 (minutes)

40 x + y = 40 x = 25

30 Self Practice 7.1b

20 1. (a) 50
(b) The car is stationary.
10 y = 10 (c) (i) 40
x (ii) The car moves for a distance of 100 km with
an average speed of 40 km h–1 in 2.5 hours.
O 10 20 30 40 (pesak)
2. (a) 2
(b) 4.8

(c) Encik Rashid runs for a distance of 4Skamizwisthebenar
an average speed of 4.8 km h–1 in 50 minutes.

303

3. (a) 1424 Self Practice 7.2c
(b) (i) The car is stationary for 66 minutes.
(ii) The car moves with an average speed of 1. (a) The motorcycle is decelerating at 0.75 m s–2 in
40 km h–1 for a distance of 30 km in
45 minutes. 20 seconds; or the speed of the motorcycle

4. (a) 40 decreases from 35 m s–1 to 20 m s–1 in 20 seconds;
(b) The car moves with an average speed of 54 km h–1
or the motorcycle moves 550 m in 20 seconds.
for a distance of 36 km in 40 minutes.
(b) The motorcycle moves at a uniform speed of

20 m s–1 in 30 seconds; or the motorcycle travels

for 600 m at a uniform speed.

Self Practice 7.1c 2. (a) 5 m s–2 (b) 260 m
6
1. (a) 3
(c) The particle moves at a uniform speed of 15 m s–1
b es, effrey will finish the race in seconds
in 7 seconds.
2. (a) 50 (b) 70 3
3. (a) 8 m s–2 (b) 1 200 m
(c) (d) 11:12 in the morning

Distance (km) (c) Encik Merisat drives for a distance of 1.725 km in

100 2.5 minutes with an average speed of 41.4 km h–1.

60 Self Practice 7.2d

Time 1. (a) 96 (b) 18 (c) 14
O 50 70 102 (minutes) 2. (a) 1 (b) 25.5
3. (a) 28 (b) 15
3. (a) 25 minutes

(b) (i) 27 (ii) 33 km

(c) 80

(d) 45 1. (a) 6 minutes (b) 60 (c) 42.86

4. (a) 20 2. (a) 100 (b) 1.6 (c) 57.14

(b) 60 3. (a) 8 seconds (b) 17

(c) The car moves with an average speed of 72 km h–1 4. (a) – 7 (b) 6 (c) 19.68 km h–1
6
for a distance of 36 km in 30 minutes.

5. (a) 12 (b) 32.4 (c) 60

Self Practice 7.2a 6. (a) 80 (b) 0915 hours

1. (a) Speed (m s–1) 7. (a) (i) 80

(b) Speed (km min–1) (ii) Car A moves with an average speed of

8 40 25 m s–1 for a distance of 2 km in 80 seconds.
6
4 30 (b) 1 minute
2
Time 20 CHAPTER 8 Measures of Dispersion for
O 246 (seconds) Ungrouped Data
10
Time

O 1 2 3 4 (minutes)

2. (a) Time, t (seconds) 0 30 (b) Time, t (seconds) 0 5 Self Practice 8.1a (b) 105

Speed, v (m s–1) 60 0 Speed, v (m s–1) 0 15 1 (a) 45, 150
2. p = 30, q = 120
Speed (m s–1) Speed (m s–1) 3. 2.3
60 15
40 Time Self Practice 8.1b Group B
20 (seconds) 10
1. Group A
O 123 5
Time 976541 4 00122689
86443220 5 224566778899
O 2 4 6 (seconds) 9887643322100 6 013445699
66532000 7 0025668
Self Practice 7.2b
64221 8 2334
1. (a) 360 (b) 0.275 (c) 2.6
2. (a) 1432 (b) 2931
3. (a) 16 (b) 22.5 In general, the body mass of pupils in group A is
greater than the body mass of pupils in group B.

304

Index

Acceleration 195, 200, 203 Financial plan 272, 274, 277 Range 212, 219, 224
Active income 274, 276 Financial planning 272, 273, Rate of change in distance 184,
Antecedent 63, 64, 68 277, 279 189, 191
Argument 71, 72, 75 Fixed expenses 274, 278, 282 Rate of change of speed 195,
Assets 274 Frequency table 212, 219, 221 200, 201
Attainable 273, 274, 283 Realistic 273, 274, 283
Average speed 189, 197, 203 Graph 130, 135, 139 Region 158
Axis of symmetry 8, 9, 12 Root 16, 17, 18, 22
Implication 63, 66, 68
Bankruptcy 289 Independent events 246, 249 Sample space 244, 245
Box plot 226, 227, 230 Inductive 72, 73, 81 Short-term 272, 274, 282, 285
Inflation 284, 285, 288 Simple graph 131, 134, 139
Cash flow 274, 276, 282 Interquartile range 212, 219, SMART 273, 279, 280, 283
Coefficient 4 220 Solid line 158
Combined events 244, 246 Intersection of sets 96, 100 Specific 273, 277, 283
Common region 169, 174 Inverse 66, 67, 68 Speed 184, 187, 189
Complement 100, 110 Speed-time graph 195, 197
Compound statement 60, 89 Liabilities 274 Standard deviation 212, 221,
Conclusion 71, 73, 75 Linear inequality 156, 158 224
Consequent 63, 64, 68 Linear inequality system 167 Statement 56, 58, 59, 60
Constant 3, 23 Long-term 272, 274, 275, 282 Stationary 187, 194
Contrapositive 66, 67, 68 Loop 132, 136 Stem-and-leaf plot 214, 216
Converse 66, 67, 68 Subgraph 139
Counter-example 69, 70 Maximum point 7, 8, 9
Cumulative frequency 221 Measurable 273, 283 Time 184, 195
Median 220, 226, 227 Time-bound 273, 274, 283
Dashed line 158, 162, 171 Minimum point 7, 8, 9 Tree 139, 146
Deceleration 200, 202 Multiple edges 131, 132 Truth value 56, 57, 59
Deductive 71, 73, 75 Mutually exclusive events 253
Degree 131, 133, 136 Undirected graph 135, 143
Dependent events 246, 247 Needs 273, 280 Ungrouped data 219, 220
Digit value 36, 37, 39 Negation 59, 70 Uniform speed 187, 200
Directed graph 135, 136 Network 130, 143 Union of sets 106, 110
Dispersion 212, 213, 217 Non-mutually exclusive events Unweighted graph 137, 143
Distance 184, 197 253, 255, 259
Distance-time graph 184, 197 Number base 34, 45 Validity 75
Dot plot 213, 215, 217 Variable 2, 3, 16, 156
Observation 213 Variable expenses 274, 278
Edge 130, 132, 135 Outlier 224, 229 Variance 221, 228, 229
Expenses 272, 273, 274 Vertex 130, 132, 139
Expression 2, 3, 16 Passive income 274, 276, 287
Extreme value 224, 228 Place value 35, 36, 39 Wants 273
Premise 71, 72, 81 Weighted graph 137, 143
Factorisation 21, 23
Financial goal 272, 273, 274, Quadratic 2, 3, 7, 9
275, 282 Quadratic equation 15, 16, 21
Quadratic function 2, 5, 7, 9

312