NCERT Textbook - Motion in a Straight Line Class 11 Notes | EduRev

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Class 11 : NCERT Textbook - Motion in a Straight Line Class 11 Notes | EduRev

 Page 1


CHAPTER THREE
MOTION IN A STRAIGHT LINE
3.1 Introduction
3.2 Position, path length and
displacement
3.3 Average velocity and average
speed
3.4 Instantaneous velocity and
speed
3.5 Acceleration
3.6 Kinematic equations for
uniformly accelerated motion
3.7 Relative velocity
Summary
Points to ponder
Exercises
Additional exercises
Appendix 3.1
3.1  INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle.  Even when we are sleeping, air moves
into and out of our lungs and blood flows in arteries and
veins.  We see leaves falling from trees and water flowing
down a dam.  Automobiles and planes carry people from one
place to the other. The earth rotates once every twenty-four
hours and revolves round the sun once in a year. The sun
itself is in motion in the Milky Way, which is again moving
within its local group of galaxies.
Motion is change in position of an object with time. How
does the position change with time ? In this chapter, we shall
learn how to describe motion. For this, we develop the
concepts of velocity and acceleration. We shall confine
ourselves to the study of motion of objects along a straight
line, also known as rectilinear motion. For the case of
rectilinear motion with uniform acceleration, a set of simple
equations can be obtained. Finally, to understand the relative
nature of motion, we introduce the concept of relative velocity.
In our discussions, we shall treat the objects in motion as
point objects. This approximation is valid so far as the size
of the object is much smaller than the distance it moves in a
reasonable duration of time.  In a good number of situations
in real-life, the size of objects can be neglected and they can
be considered as point-like objects without much error.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 5.
3.2  POSITION, PATH LENGTH AND DISPLACEMENT
Earlier you learnt that motion is change in position of an
object with time. In order to specify position, we need to use
a reference point and a set of axes. It is convenient to choose
© NCERT
not to be republished
Page 2


CHAPTER THREE
MOTION IN A STRAIGHT LINE
3.1 Introduction
3.2 Position, path length and
displacement
3.3 Average velocity and average
speed
3.4 Instantaneous velocity and
speed
3.5 Acceleration
3.6 Kinematic equations for
uniformly accelerated motion
3.7 Relative velocity
Summary
Points to ponder
Exercises
Additional exercises
Appendix 3.1
3.1  INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle.  Even when we are sleeping, air moves
into and out of our lungs and blood flows in arteries and
veins.  We see leaves falling from trees and water flowing
down a dam.  Automobiles and planes carry people from one
place to the other. The earth rotates once every twenty-four
hours and revolves round the sun once in a year. The sun
itself is in motion in the Milky Way, which is again moving
within its local group of galaxies.
Motion is change in position of an object with time. How
does the position change with time ? In this chapter, we shall
learn how to describe motion. For this, we develop the
concepts of velocity and acceleration. We shall confine
ourselves to the study of motion of objects along a straight
line, also known as rectilinear motion. For the case of
rectilinear motion with uniform acceleration, a set of simple
equations can be obtained. Finally, to understand the relative
nature of motion, we introduce the concept of relative velocity.
In our discussions, we shall treat the objects in motion as
point objects. This approximation is valid so far as the size
of the object is much smaller than the distance it moves in a
reasonable duration of time.  In a good number of situations
in real-life, the size of objects can be neglected and they can
be considered as point-like objects without much error.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 5.
3.2  POSITION, PATH LENGTH AND DISPLACEMENT
Earlier you learnt that motion is change in position of an
object with time. In order to specify position, we need to use
a reference point and a set of axes. It is convenient to choose
© NCERT
not to be republished
PHYSICS 40
with the path of the car?s motion and origin of
the axis as the point from where the car started
moving, i.e. the car was at x = 0 at t = 0 (Fig. 3.1).
Let P, Q and R represent the positions of the car
at different instants of time.  Consider two cases
of motion.  In the first case, the car moves from
O to P.  Then the distance moved by the car is
OP = +360 m. This distance is called the path
length  traversed by the car.   In the second
case, the car moves from O to P and then moves
back from P to Q.   During this course of motion,
the path length traversed is OP + PQ = + 360 m
+ (+120 m) = + 480 m.  Path length is a scalar
quantity ? a quantity that has a magnitude
only and no direction (see Chapter 4).
Displacement
It is useful to define another quantity
displacement  as  the  change  in position. Let
x
1
 and x
2
 
be the positions of an object at time t
1
and t
2
. Then its displacement, denoted by ?x, in
time ?t = (t
2 
- t
1
), is given by the difference
between the final and initial positions :
                       ?x = x
2
 
? x
1
(We use the Greek letter delta (?) to denote a
change in a quantity.)
If   x
2
 > x
1
, ?x is positive; and if x
2
 
< x
1
,
 
?x is
negative.
Displacement has both magnitude and
direction. Such quantities are represented by
vectors. You will read about vectors in the next
chapter. Presently, we are dealing with motion
along a straight line (also called rectilinear
motion) only. In one-dimensional motion, there
are only two directions (backward and forward,
upward and downward) in which an object can
move, and these two directions can easily be
specified by + and ? signs. For example,
displacement of the car in moving from O to P is :
?x = x
2
 ? x
1
 = (+360 m) ? 0 m = +360 m
The displacement has a magnitude of 360 m and
is directed in the positive x direction as indicated
by the + sign. Similarly, the displacement of the
car from P to Q is 240 m ? 360 m = ? 120 m. The
Fig. 3.1  x-axis, origin and positions of a car at different times.
a rectangular coordinate system consisting of
three mutually perpenducular axes, labelled  X-,
Y-, and Z- axes. The point of intersection of these
three axes is called origin (O) and serves as the
reference point. The coordinates (x, y. z) of an
object describe the position of the object with
respect to this coordinate system. To measure
time, we position a clock in this system. This
coordinate system along with a clock constitutes
a frame of reference.
If one or more coordinates of an object change
with time, we say that the object is in motion.
Otherwise, the object is said to be at rest with
respect to this frame of reference.
The choice of a set of axes in a frame of
reference depends upon the situation. For
example, for describing motion in one dimension,
we need only one axis. To describe motion in
two/three dimensions, we need a set of two/
three axes.
Description of an event depends on the frame
of reference chosen for the description. For
example, when you say that a car is moving on
a road, you are describing the car with respect
to a frame of reference attached to you or to the
ground. But with respect to a frame of reference
attached with a person sitting in the car, the
car is at rest.
To describe motion along a straight line, we
can choose an axis, say X-axis, so that it
coincides with the path of the object. We then
measure the position of the object with reference
to a conveniently chosen origin, say O, as shown
in Fig. 3.1. Positions to the right of O are taken
as positive and to the left of O, as negative.
Following this convention, the position
coordinates of point P and Q in Fig. 3.1 are +360
m and +240 m. Similarly, the position coordinate
of point R is ?120 m.
Path length
Consider the motion of a car along a straight
line.  We choose the x-axis such that it coincides
© NCERT
not to be republished
Page 3


CHAPTER THREE
MOTION IN A STRAIGHT LINE
3.1 Introduction
3.2 Position, path length and
displacement
3.3 Average velocity and average
speed
3.4 Instantaneous velocity and
speed
3.5 Acceleration
3.6 Kinematic equations for
uniformly accelerated motion
3.7 Relative velocity
Summary
Points to ponder
Exercises
Additional exercises
Appendix 3.1
3.1  INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle.  Even when we are sleeping, air moves
into and out of our lungs and blood flows in arteries and
veins.  We see leaves falling from trees and water flowing
down a dam.  Automobiles and planes carry people from one
place to the other. The earth rotates once every twenty-four
hours and revolves round the sun once in a year. The sun
itself is in motion in the Milky Way, which is again moving
within its local group of galaxies.
Motion is change in position of an object with time. How
does the position change with time ? In this chapter, we shall
learn how to describe motion. For this, we develop the
concepts of velocity and acceleration. We shall confine
ourselves to the study of motion of objects along a straight
line, also known as rectilinear motion. For the case of
rectilinear motion with uniform acceleration, a set of simple
equations can be obtained. Finally, to understand the relative
nature of motion, we introduce the concept of relative velocity.
In our discussions, we shall treat the objects in motion as
point objects. This approximation is valid so far as the size
of the object is much smaller than the distance it moves in a
reasonable duration of time.  In a good number of situations
in real-life, the size of objects can be neglected and they can
be considered as point-like objects without much error.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 5.
3.2  POSITION, PATH LENGTH AND DISPLACEMENT
Earlier you learnt that motion is change in position of an
object with time. In order to specify position, we need to use
a reference point and a set of axes. It is convenient to choose
© NCERT
not to be republished
PHYSICS 40
with the path of the car?s motion and origin of
the axis as the point from where the car started
moving, i.e. the car was at x = 0 at t = 0 (Fig. 3.1).
Let P, Q and R represent the positions of the car
at different instants of time.  Consider two cases
of motion.  In the first case, the car moves from
O to P.  Then the distance moved by the car is
OP = +360 m. This distance is called the path
length  traversed by the car.   In the second
case, the car moves from O to P and then moves
back from P to Q.   During this course of motion,
the path length traversed is OP + PQ = + 360 m
+ (+120 m) = + 480 m.  Path length is a scalar
quantity ? a quantity that has a magnitude
only and no direction (see Chapter 4).
Displacement
It is useful to define another quantity
displacement  as  the  change  in position. Let
x
1
 and x
2
 
be the positions of an object at time t
1
and t
2
. Then its displacement, denoted by ?x, in
time ?t = (t
2 
- t
1
), is given by the difference
between the final and initial positions :
                       ?x = x
2
 
? x
1
(We use the Greek letter delta (?) to denote a
change in a quantity.)
If   x
2
 > x
1
, ?x is positive; and if x
2
 
< x
1
,
 
?x is
negative.
Displacement has both magnitude and
direction. Such quantities are represented by
vectors. You will read about vectors in the next
chapter. Presently, we are dealing with motion
along a straight line (also called rectilinear
motion) only. In one-dimensional motion, there
are only two directions (backward and forward,
upward and downward) in which an object can
move, and these two directions can easily be
specified by + and ? signs. For example,
displacement of the car in moving from O to P is :
?x = x
2
 ? x
1
 = (+360 m) ? 0 m = +360 m
The displacement has a magnitude of 360 m and
is directed in the positive x direction as indicated
by the + sign. Similarly, the displacement of the
car from P to Q is 240 m ? 360 m = ? 120 m. The
Fig. 3.1  x-axis, origin and positions of a car at different times.
a rectangular coordinate system consisting of
three mutually perpenducular axes, labelled  X-,
Y-, and Z- axes. The point of intersection of these
three axes is called origin (O) and serves as the
reference point. The coordinates (x, y. z) of an
object describe the position of the object with
respect to this coordinate system. To measure
time, we position a clock in this system. This
coordinate system along with a clock constitutes
a frame of reference.
If one or more coordinates of an object change
with time, we say that the object is in motion.
Otherwise, the object is said to be at rest with
respect to this frame of reference.
The choice of a set of axes in a frame of
reference depends upon the situation. For
example, for describing motion in one dimension,
we need only one axis. To describe motion in
two/three dimensions, we need a set of two/
three axes.
Description of an event depends on the frame
of reference chosen for the description. For
example, when you say that a car is moving on
a road, you are describing the car with respect
to a frame of reference attached to you or to the
ground. But with respect to a frame of reference
attached with a person sitting in the car, the
car is at rest.
To describe motion along a straight line, we
can choose an axis, say X-axis, so that it
coincides with the path of the object. We then
measure the position of the object with reference
to a conveniently chosen origin, say O, as shown
in Fig. 3.1. Positions to the right of O are taken
as positive and to the left of O, as negative.
Following this convention, the position
coordinates of point P and Q in Fig. 3.1 are +360
m and +240 m. Similarly, the position coordinate
of point R is ?120 m.
Path length
Consider the motion of a car along a straight
line.  We choose the x-axis such that it coincides
© NCERT
not to be republished
MOTION IN A STRAIGHT LINE 41
negative sign indicates the direction of
displacement. Thus, it is not necessary to use
vector notation for discussing motion of objects
in one-dimension.
The magnitude of displacement may or may
not be equal to the path length traversed by
an object.  For example, for motion of the car
from O to P, the path length is  +360 m and the
displacement is +360 m. In this case, the
magnitude of displacement (360 m) is equal to
the path length (360 m). But consider the motion
of the car from O to P and back to Q. In this
case, the path length = (+360 m) + (+120 m) = +
480 m. However, the displacement = (+240 m) ?
(0 m) =  + 240 m.  Thus, the magnitude of
displacement (240 m) is not equal to the path
length (480 m).
The magnitude of the displacement for a
course of motion may be zero but the
corresponding path length is not zero.  For
example, if the car starts from O, goes to P and
then returns to O, the final position coincides
with the initial position and the displacement
is zero. However, the path length of this journey
is OP + PO = 360 m +  360 m = 720 m.
Motion of an object can be represented by a
position-time graph as you have already learnt
about it. Such a graph is a powerful tool to
represent and analyse different aspects of
motion of an object.  For motion along a straight
line, say X-axis, only x-coordinate varies with
time and we have an x-t graph. Let us first
consider the simple case in which an object is
stationary, e.g. a car standing still at x = 40 m.
The position-time graph is a straight line parallel
to the time axis, as shown in Fig. 3.2(a).
If an object moving along the straight line
covers equal distances in equal intervals of
time, it is said to be in uniform motion along a
straight line. Fig. 3.2(b) shows the position-time
graph of such a motion.
Fig. 3.2  Position-time graph of (a) stationary object, and (b) an object in uniform motion.
Fig. 3.3  Position-time graph of a car.
t (s) !
"
x
(m)
© NCERT
not to be republished
Page 4


CHAPTER THREE
MOTION IN A STRAIGHT LINE
3.1 Introduction
3.2 Position, path length and
displacement
3.3 Average velocity and average
speed
3.4 Instantaneous velocity and
speed
3.5 Acceleration
3.6 Kinematic equations for
uniformly accelerated motion
3.7 Relative velocity
Summary
Points to ponder
Exercises
Additional exercises
Appendix 3.1
3.1  INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle.  Even when we are sleeping, air moves
into and out of our lungs and blood flows in arteries and
veins.  We see leaves falling from trees and water flowing
down a dam.  Automobiles and planes carry people from one
place to the other. The earth rotates once every twenty-four
hours and revolves round the sun once in a year. The sun
itself is in motion in the Milky Way, which is again moving
within its local group of galaxies.
Motion is change in position of an object with time. How
does the position change with time ? In this chapter, we shall
learn how to describe motion. For this, we develop the
concepts of velocity and acceleration. We shall confine
ourselves to the study of motion of objects along a straight
line, also known as rectilinear motion. For the case of
rectilinear motion with uniform acceleration, a set of simple
equations can be obtained. Finally, to understand the relative
nature of motion, we introduce the concept of relative velocity.
In our discussions, we shall treat the objects in motion as
point objects. This approximation is valid so far as the size
of the object is much smaller than the distance it moves in a
reasonable duration of time.  In a good number of situations
in real-life, the size of objects can be neglected and they can
be considered as point-like objects without much error.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 5.
3.2  POSITION, PATH LENGTH AND DISPLACEMENT
Earlier you learnt that motion is change in position of an
object with time. In order to specify position, we need to use
a reference point and a set of axes. It is convenient to choose
© NCERT
not to be republished
PHYSICS 40
with the path of the car?s motion and origin of
the axis as the point from where the car started
moving, i.e. the car was at x = 0 at t = 0 (Fig. 3.1).
Let P, Q and R represent the positions of the car
at different instants of time.  Consider two cases
of motion.  In the first case, the car moves from
O to P.  Then the distance moved by the car is
OP = +360 m. This distance is called the path
length  traversed by the car.   In the second
case, the car moves from O to P and then moves
back from P to Q.   During this course of motion,
the path length traversed is OP + PQ = + 360 m
+ (+120 m) = + 480 m.  Path length is a scalar
quantity ? a quantity that has a magnitude
only and no direction (see Chapter 4).
Displacement
It is useful to define another quantity
displacement  as  the  change  in position. Let
x
1
 and x
2
 
be the positions of an object at time t
1
and t
2
. Then its displacement, denoted by ?x, in
time ?t = (t
2 
- t
1
), is given by the difference
between the final and initial positions :
                       ?x = x
2
 
? x
1
(We use the Greek letter delta (?) to denote a
change in a quantity.)
If   x
2
 > x
1
, ?x is positive; and if x
2
 
< x
1
,
 
?x is
negative.
Displacement has both magnitude and
direction. Such quantities are represented by
vectors. You will read about vectors in the next
chapter. Presently, we are dealing with motion
along a straight line (also called rectilinear
motion) only. In one-dimensional motion, there
are only two directions (backward and forward,
upward and downward) in which an object can
move, and these two directions can easily be
specified by + and ? signs. For example,
displacement of the car in moving from O to P is :
?x = x
2
 ? x
1
 = (+360 m) ? 0 m = +360 m
The displacement has a magnitude of 360 m and
is directed in the positive x direction as indicated
by the + sign. Similarly, the displacement of the
car from P to Q is 240 m ? 360 m = ? 120 m. The
Fig. 3.1  x-axis, origin and positions of a car at different times.
a rectangular coordinate system consisting of
three mutually perpenducular axes, labelled  X-,
Y-, and Z- axes. The point of intersection of these
three axes is called origin (O) and serves as the
reference point. The coordinates (x, y. z) of an
object describe the position of the object with
respect to this coordinate system. To measure
time, we position a clock in this system. This
coordinate system along with a clock constitutes
a frame of reference.
If one or more coordinates of an object change
with time, we say that the object is in motion.
Otherwise, the object is said to be at rest with
respect to this frame of reference.
The choice of a set of axes in a frame of
reference depends upon the situation. For
example, for describing motion in one dimension,
we need only one axis. To describe motion in
two/three dimensions, we need a set of two/
three axes.
Description of an event depends on the frame
of reference chosen for the description. For
example, when you say that a car is moving on
a road, you are describing the car with respect
to a frame of reference attached to you or to the
ground. But with respect to a frame of reference
attached with a person sitting in the car, the
car is at rest.
To describe motion along a straight line, we
can choose an axis, say X-axis, so that it
coincides with the path of the object. We then
measure the position of the object with reference
to a conveniently chosen origin, say O, as shown
in Fig. 3.1. Positions to the right of O are taken
as positive and to the left of O, as negative.
Following this convention, the position
coordinates of point P and Q in Fig. 3.1 are +360
m and +240 m. Similarly, the position coordinate
of point R is ?120 m.
Path length
Consider the motion of a car along a straight
line.  We choose the x-axis such that it coincides
© NCERT
not to be republished
MOTION IN A STRAIGHT LINE 41
negative sign indicates the direction of
displacement. Thus, it is not necessary to use
vector notation for discussing motion of objects
in one-dimension.
The magnitude of displacement may or may
not be equal to the path length traversed by
an object.  For example, for motion of the car
from O to P, the path length is  +360 m and the
displacement is +360 m. In this case, the
magnitude of displacement (360 m) is equal to
the path length (360 m). But consider the motion
of the car from O to P and back to Q. In this
case, the path length = (+360 m) + (+120 m) = +
480 m. However, the displacement = (+240 m) ?
(0 m) =  + 240 m.  Thus, the magnitude of
displacement (240 m) is not equal to the path
length (480 m).
The magnitude of the displacement for a
course of motion may be zero but the
corresponding path length is not zero.  For
example, if the car starts from O, goes to P and
then returns to O, the final position coincides
with the initial position and the displacement
is zero. However, the path length of this journey
is OP + PO = 360 m +  360 m = 720 m.
Motion of an object can be represented by a
position-time graph as you have already learnt
about it. Such a graph is a powerful tool to
represent and analyse different aspects of
motion of an object.  For motion along a straight
line, say X-axis, only x-coordinate varies with
time and we have an x-t graph. Let us first
consider the simple case in which an object is
stationary, e.g. a car standing still at x = 40 m.
The position-time graph is a straight line parallel
to the time axis, as shown in Fig. 3.2(a).
If an object moving along the straight line
covers equal distances in equal intervals of
time, it is said to be in uniform motion along a
straight line. Fig. 3.2(b) shows the position-time
graph of such a motion.
Fig. 3.2  Position-time graph of (a) stationary object, and (b) an object in uniform motion.
Fig. 3.3  Position-time graph of a car.
t (s) !
"
x
(m)
© NCERT
not to be republished
PHYSICS 42
Now, let us consider the motion of a car that
starts from rest at time t = 0 s from the origin O
and picks up speed till t = 10 s and thereafter
moves with uniform speed till t = 18 s. Then the
brakes are applied and the car stops at
t = 20 s and x = 296 m. The position-time graph
for this case is shown in Fig. 3.3. We shall refer
to this graph in our discussion in the following
sections.
3.3  AVERAGE VELOCITY AND AVERAGE
SPEED
When an object is in motion, its position
changes with time.  But how fast is the position
changing with time and in what direction?  To
describe this, we define the quantity average
velocity. Average velocity is defined as the
change in position or displacement (?x) divided
by the time intervals (?t), in which the
displacement occurs :
     
v
xx
tt
x
t
21
21
=
-
-
=
?
?
(3.1)
where x
2
  and x
1
 are the positions of the object
at time t
2
and  t
1
, respectively. Here the bar over
the symbol for velocity is a standard notation
used to indicate an average quantity.  The SI
unit for velocity is m/s or m s
?1
, although km h
?1
is used in many everyday applications.
Like displacement, average velocity is also a
vector quantity. But as explained earlier, for
motion in a straight line, the directional aspect
of the vector can be taken care of by + and ?
signs and we do not have to use the vector
notation for velocity in this chapter.
Fig. 3.4 The average velocity is the slope of line P
1
P
2
.
Consider the motion of the car in Fig. 3.3. The
portion of the x-t graph between t = 0 s and t = 8
s is blown up and shown in Fig. 3.4.  As seen
from the plot, the average velocity of the car
between time t = 5 s and t = 7 s is :
( )
()
1 –
1 2
1 2
s m  8.7
s 5 7
m 0 10 4 27
=
-
-
=
-
-
=
. .
t t
x x
v
Geometrically, this is the slope of the straight
line P
1
P
2
 connecting the initial position 
1
P to
the final position P
2
   
as
   
shown in Fig. 3.4.
The average velocity can be positive or negative
depending upon the sign of the displacement. It
is zero if the displacement is zero. Fig. 3.5 shows
the x-t graphs for an object, moving with positive
velocity (Fig. 3.5a), moving with negative velocity
(Fig. 3.5b)  and at rest (Fig. 3.5c).
Average velocity as defined above involves
only the displacement of the object. We have seen
earlier that the magnitude of displacement may
be different from the actual path length. To
describe the rate of motion over the actual path,
we introduce another quantity called average
speed.
Average speed  is defined as the total path
length travelled divided by the total time
interval during which the motion has taken
place :
Average speed
Total path length
Total time interval 
=
  (3.2)
Average speed has obviously the same unit
(m s
?1
) as that of velocity.  But it does not tell us
in what direction an object is moving.  Thus, it
is always positive (in contrast to the average
velocity which can be positive or negative). If the
motion of an object is along a straight line and
in the same direction, the magnitude of
displacement is equal to the total path length.
In that case, the magnitude of average velocity
Fig. 3.5 Position-time graph for an object (a) moving
with positive velocity, (b) moving with
negative velocity, and (c) at rest.
© NCERT
not to be republished
Page 5


CHAPTER THREE
MOTION IN A STRAIGHT LINE
3.1 Introduction
3.2 Position, path length and
displacement
3.3 Average velocity and average
speed
3.4 Instantaneous velocity and
speed
3.5 Acceleration
3.6 Kinematic equations for
uniformly accelerated motion
3.7 Relative velocity
Summary
Points to ponder
Exercises
Additional exercises
Appendix 3.1
3.1  INTRODUCTION
Motion is common to everything in the universe. We walk,
run and ride a bicycle.  Even when we are sleeping, air moves
into and out of our lungs and blood flows in arteries and
veins.  We see leaves falling from trees and water flowing
down a dam.  Automobiles and planes carry people from one
place to the other. The earth rotates once every twenty-four
hours and revolves round the sun once in a year. The sun
itself is in motion in the Milky Way, which is again moving
within its local group of galaxies.
Motion is change in position of an object with time. How
does the position change with time ? In this chapter, we shall
learn how to describe motion. For this, we develop the
concepts of velocity and acceleration. We shall confine
ourselves to the study of motion of objects along a straight
line, also known as rectilinear motion. For the case of
rectilinear motion with uniform acceleration, a set of simple
equations can be obtained. Finally, to understand the relative
nature of motion, we introduce the concept of relative velocity.
In our discussions, we shall treat the objects in motion as
point objects. This approximation is valid so far as the size
of the object is much smaller than the distance it moves in a
reasonable duration of time.  In a good number of situations
in real-life, the size of objects can be neglected and they can
be considered as point-like objects without much error.
In Kinematics, we study ways to describe motion without
going into the causes of motion. What causes motion
described in this chapter and the next chapter forms the
subject matter of Chapter 5.
3.2  POSITION, PATH LENGTH AND DISPLACEMENT
Earlier you learnt that motion is change in position of an
object with time. In order to specify position, we need to use
a reference point and a set of axes. It is convenient to choose
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PHYSICS 40
with the path of the car?s motion and origin of
the axis as the point from where the car started
moving, i.e. the car was at x = 0 at t = 0 (Fig. 3.1).
Let P, Q and R represent the positions of the car
at different instants of time.  Consider two cases
of motion.  In the first case, the car moves from
O to P.  Then the distance moved by the car is
OP = +360 m. This distance is called the path
length  traversed by the car.   In the second
case, the car moves from O to P and then moves
back from P to Q.   During this course of motion,
the path length traversed is OP + PQ = + 360 m
+ (+120 m) = + 480 m.  Path length is a scalar
quantity ? a quantity that has a magnitude
only and no direction (see Chapter 4).
Displacement
It is useful to define another quantity
displacement  as  the  change  in position. Let
x
1
 and x
2
 
be the positions of an object at time t
1
and t
2
. Then its displacement, denoted by ?x, in
time ?t = (t
2 
- t
1
), is given by the difference
between the final and initial positions :
                       ?x = x
2
 
? x
1
(We use the Greek letter delta (?) to denote a
change in a quantity.)
If   x
2
 > x
1
, ?x is positive; and if x
2
 
< x
1
,
 
?x is
negative.
Displacement has both magnitude and
direction. Such quantities are represented by
vectors. You will read about vectors in the next
chapter. Presently, we are dealing with motion
along a straight line (also called rectilinear
motion) only. In one-dimensional motion, there
are only two directions (backward and forward,
upward and downward) in which an object can
move, and these two directions can easily be
specified by + and ? signs. For example,
displacement of the car in moving from O to P is :
?x = x
2
 ? x
1
 = (+360 m) ? 0 m = +360 m
The displacement has a magnitude of 360 m and
is directed in the positive x direction as indicated
by the + sign. Similarly, the displacement of the
car from P to Q is 240 m ? 360 m = ? 120 m. The
Fig. 3.1  x-axis, origin and positions of a car at different times.
a rectangular coordinate system consisting of
three mutually perpenducular axes, labelled  X-,
Y-, and Z- axes. The point of intersection of these
three axes is called origin (O) and serves as the
reference point. The coordinates (x, y. z) of an
object describe the position of the object with
respect to this coordinate system. To measure
time, we position a clock in this system. This
coordinate system along with a clock constitutes
a frame of reference.
If one or more coordinates of an object change
with time, we say that the object is in motion.
Otherwise, the object is said to be at rest with
respect to this frame of reference.
The choice of a set of axes in a frame of
reference depends upon the situation. For
example, for describing motion in one dimension,
we need only one axis. To describe motion in
two/three dimensions, we need a set of two/
three axes.
Description of an event depends on the frame
of reference chosen for the description. For
example, when you say that a car is moving on
a road, you are describing the car with respect
to a frame of reference attached to you or to the
ground. But with respect to a frame of reference
attached with a person sitting in the car, the
car is at rest.
To describe motion along a straight line, we
can choose an axis, say X-axis, so that it
coincides with the path of the object. We then
measure the position of the object with reference
to a conveniently chosen origin, say O, as shown
in Fig. 3.1. Positions to the right of O are taken
as positive and to the left of O, as negative.
Following this convention, the position
coordinates of point P and Q in Fig. 3.1 are +360
m and +240 m. Similarly, the position coordinate
of point R is ?120 m.
Path length
Consider the motion of a car along a straight
line.  We choose the x-axis such that it coincides
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MOTION IN A STRAIGHT LINE 41
negative sign indicates the direction of
displacement. Thus, it is not necessary to use
vector notation for discussing motion of objects
in one-dimension.
The magnitude of displacement may or may
not be equal to the path length traversed by
an object.  For example, for motion of the car
from O to P, the path length is  +360 m and the
displacement is +360 m. In this case, the
magnitude of displacement (360 m) is equal to
the path length (360 m). But consider the motion
of the car from O to P and back to Q. In this
case, the path length = (+360 m) + (+120 m) = +
480 m. However, the displacement = (+240 m) ?
(0 m) =  + 240 m.  Thus, the magnitude of
displacement (240 m) is not equal to the path
length (480 m).
The magnitude of the displacement for a
course of motion may be zero but the
corresponding path length is not zero.  For
example, if the car starts from O, goes to P and
then returns to O, the final position coincides
with the initial position and the displacement
is zero. However, the path length of this journey
is OP + PO = 360 m +  360 m = 720 m.
Motion of an object can be represented by a
position-time graph as you have already learnt
about it. Such a graph is a powerful tool to
represent and analyse different aspects of
motion of an object.  For motion along a straight
line, say X-axis, only x-coordinate varies with
time and we have an x-t graph. Let us first
consider the simple case in which an object is
stationary, e.g. a car standing still at x = 40 m.
The position-time graph is a straight line parallel
to the time axis, as shown in Fig. 3.2(a).
If an object moving along the straight line
covers equal distances in equal intervals of
time, it is said to be in uniform motion along a
straight line. Fig. 3.2(b) shows the position-time
graph of such a motion.
Fig. 3.2  Position-time graph of (a) stationary object, and (b) an object in uniform motion.
Fig. 3.3  Position-time graph of a car.
t (s) !
"
x
(m)
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PHYSICS 42
Now, let us consider the motion of a car that
starts from rest at time t = 0 s from the origin O
and picks up speed till t = 10 s and thereafter
moves with uniform speed till t = 18 s. Then the
brakes are applied and the car stops at
t = 20 s and x = 296 m. The position-time graph
for this case is shown in Fig. 3.3. We shall refer
to this graph in our discussion in the following
sections.
3.3  AVERAGE VELOCITY AND AVERAGE
SPEED
When an object is in motion, its position
changes with time.  But how fast is the position
changing with time and in what direction?  To
describe this, we define the quantity average
velocity. Average velocity is defined as the
change in position or displacement (?x) divided
by the time intervals (?t), in which the
displacement occurs :
     
v
xx
tt
x
t
21
21
=
-
-
=
?
?
(3.1)
where x
2
  and x
1
 are the positions of the object
at time t
2
and  t
1
, respectively. Here the bar over
the symbol for velocity is a standard notation
used to indicate an average quantity.  The SI
unit for velocity is m/s or m s
?1
, although km h
?1
is used in many everyday applications.
Like displacement, average velocity is also a
vector quantity. But as explained earlier, for
motion in a straight line, the directional aspect
of the vector can be taken care of by + and ?
signs and we do not have to use the vector
notation for velocity in this chapter.
Fig. 3.4 The average velocity is the slope of line P
1
P
2
.
Consider the motion of the car in Fig. 3.3. The
portion of the x-t graph between t = 0 s and t = 8
s is blown up and shown in Fig. 3.4.  As seen
from the plot, the average velocity of the car
between time t = 5 s and t = 7 s is :
( )
()
1 –
1 2
1 2
s m  8.7
s 5 7
m 0 10 4 27
=
-
-
=
-
-
=
. .
t t
x x
v
Geometrically, this is the slope of the straight
line P
1
P
2
 connecting the initial position 
1
P to
the final position P
2
   
as
   
shown in Fig. 3.4.
The average velocity can be positive or negative
depending upon the sign of the displacement. It
is zero if the displacement is zero. Fig. 3.5 shows
the x-t graphs for an object, moving with positive
velocity (Fig. 3.5a), moving with negative velocity
(Fig. 3.5b)  and at rest (Fig. 3.5c).
Average velocity as defined above involves
only the displacement of the object. We have seen
earlier that the magnitude of displacement may
be different from the actual path length. To
describe the rate of motion over the actual path,
we introduce another quantity called average
speed.
Average speed  is defined as the total path
length travelled divided by the total time
interval during which the motion has taken
place :
Average speed
Total path length
Total time interval 
=
  (3.2)
Average speed has obviously the same unit
(m s
?1
) as that of velocity.  But it does not tell us
in what direction an object is moving.  Thus, it
is always positive (in contrast to the average
velocity which can be positive or negative). If the
motion of an object is along a straight line and
in the same direction, the magnitude of
displacement is equal to the total path length.
In that case, the magnitude of average velocity
Fig. 3.5 Position-time graph for an object (a) moving
with positive velocity, (b) moving with
negative velocity, and (c) at rest.
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MOTION IN A STRAIGHT LINE 43
#
is equal to the average speed.  This is not always
the case, as you will see in the following example.
Example 3.1 A car is moving along a
straight line, say OP in Fig. 3.1.  It moves
from O to P in 18 s and returns from P to Q
in 6.0 s.  What are the average velocity
and average speed of the car in going (a)
from O to P ? and (b) from O to P and back
to Q ?
Answer    (a)
 
Average velocity
Displacement
Time interval 
=
   
1
+ 360 m 
20 m s
18 s
v
 
   
     
Average speed  
Path length
Time interval
=
                
1
360 m 
= 20 m s
18 s
 
 
Thus, in this case the average speed is equal to
the magnitude of the average velocity.
(b) In this case,
()
240 m
18 6.0 s
Displacement 
Average velocity =
Time interval 
+
=
+
                            
-1
=+10 m s
OP + PQ Path length
Average speed = =
Time interval t 
 
  
-1
360+120 m
= = 20 m s
24 s
Thus, in this case the average speed is not equal
to the magnitude of the average velocity. This
happens because the motion here involves
change in direction so that the path length is
greater than the magnitude of displacement.
This shows that speed is, in general, greater
than the magnitude of the velocity. #
If the car in Example 3.1 moves from O to P
and comes back to O in the same time interval,
average speed is 20 m/s but the average velocity
is zero !
3.4  INSTANTANEOUS VELOCITY AND SPEED
The average velocity tells us how fast an object
has been moving over a given time interval but
does not tell us how fast it moves at different
instants of time during that interval.  For this,
we define instantaneous velocity or simply
velocity v at an instant t.
The velocity at an instant is defined as the
limit of the average velocity as the time interval
?t becomes infinitesimally small. In other words,
vlim
x
t
=
t  0 ? 
?
? ?
(3.3a)
  
=
d
d
x
t
(3.3b)
where the symbol 
lim
t0 ??
 stands for the operation
of taking limit as  ?t$0 of the quantity on its
right. In the language of calculus, the quantity
on the right hand side of Eq. (3.3a) is the
differential coefficient of x with respect to t and
is denoted by 
 
d
d
x
t
 (see Appendix 3.1).  It is the
rate of change of position with respect to time,
at that instant.
We can use Eq. (3.3a) for obtaining the value
of velocity at an instant either graphically or
numerically. Suppose that we want to obtain
graphically the value of velocity at time  t = 4 s
(point P) for the motion of the car represented
in Fig. 3.3. The figure has been redrawn in
Fig. 3.6 choosing different scales to facilitate the
Fig. 3.6 Determining velocity from position-time
graph.  Velocity at t = 4 s is the slope of the
tangent to the graph at that instant.
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