NCERT Textbook: Motion in a Straight Line | Physics Class 11 - NEET PDF Download

 Page 1


CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 Introduction
2.2 Instantaneous velocity and
speed
2.3 Acceleration
2.4 Kinematic equations for
uniformly accelerated motion
2.5 Relative velocity
Summary
Points to ponder
Exercises
2.1  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 4.
2024-25
Page 2


CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 Introduction
2.2 Instantaneous velocity and
speed
2.3 Acceleration
2.4 Kinematic equations for
uniformly accelerated motion
2.5 Relative velocity
Summary
Points to ponder
Exercises
2.1  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 4.
2024-25
PHYSICS 14
Table 2.1  Limiting value of 
?
?
x
t
 at t = 4 s
2.2  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,
v lim
x
t
=
t  0 ? 
?
? ?
(2.1a)
  
=
d
d
x
t
(2.1b)
where the symbol 
lim
t 0 ? ?
 stands for the operation
of taking limit as  ?tg0 of the quantity on its
right. In the language of calculus, the quantity
on the right hand side of Eq. (2.1a) is the
differential coefficient of x with respect to t and
is denoted by 
 
d
d
x
t
 (see Appendix 2.1).  It is the
rate of change of position with respect to time,
at that instant.
We can use Eq. (2.1a) 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.2.1 calculation.
Let us take ?t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the
slope of line P
1
P
2
  ( Fig. 2.1) gives the value of
average velocity over the interval   3 s to 5 s.
Fig. 2.1 Determining velocity from position-time
graph.  Velocity at t = 4 s is the slope of the
tangent to the graph at that instant.
Now, we decrease the value of ?t from 2 s to 1
s.  Then line P
1
P
2
 becomes Q
1
Q
2
   and its slope
gives the value of the average velocity over
the interval 3.5 s to 4.5 s. In the limit ?t ? 0,
the line P
1
P
2
 becomes tangent to the position-
time curve at the point P and the velocity at t
= 4 s is given by the slope of the tangent at
that point. It is difficult to show this
process graphically. But if we use
numerical method to obtain the value of
the velocity, the meaning of the limiting
process becomes clear. For the graph shown
in Fig. 2.1, x = 0.08 t
3
.  Table 2.1 gives the
value of ?x/?t calculated for ?t equal to 2.0 s,
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
4.0 s. The second and third columns give the
value of t
1
= 
 
t
t
2
-
?
?
?
?
?
?
?
 and 
t t
t
2
2
= +
?
?
?
?
?
?
?
 and the
fourth and the fifth columns give the
2024-25
Page 3


CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 Introduction
2.2 Instantaneous velocity and
speed
2.3 Acceleration
2.4 Kinematic equations for
uniformly accelerated motion
2.5 Relative velocity
Summary
Points to ponder
Exercises
2.1  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 4.
2024-25
PHYSICS 14
Table 2.1  Limiting value of 
?
?
x
t
 at t = 4 s
2.2  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,
v lim
x
t
=
t  0 ? 
?
? ?
(2.1a)
  
=
d
d
x
t
(2.1b)
where the symbol 
lim
t 0 ? ?
 stands for the operation
of taking limit as  ?tg0 of the quantity on its
right. In the language of calculus, the quantity
on the right hand side of Eq. (2.1a) is the
differential coefficient of x with respect to t and
is denoted by 
 
d
d
x
t
 (see Appendix 2.1).  It is the
rate of change of position with respect to time,
at that instant.
We can use Eq. (2.1a) 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.2.1 calculation.
Let us take ?t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the
slope of line P
1
P
2
  ( Fig. 2.1) gives the value of
average velocity over the interval   3 s to 5 s.
Fig. 2.1 Determining velocity from position-time
graph.  Velocity at t = 4 s is the slope of the
tangent to the graph at that instant.
Now, we decrease the value of ?t from 2 s to 1
s.  Then line P
1
P
2
 becomes Q
1
Q
2
   and its slope
gives the value of the average velocity over
the interval 3.5 s to 4.5 s. In the limit ?t ? 0,
the line P
1
P
2
 becomes tangent to the position-
time curve at the point P and the velocity at t
= 4 s is given by the slope of the tangent at
that point. It is difficult to show this
process graphically. But if we use
numerical method to obtain the value of
the velocity, the meaning of the limiting
process becomes clear. For the graph shown
in Fig. 2.1, x = 0.08 t
3
.  Table 2.1 gives the
value of ?x/?t calculated for ?t equal to 2.0 s,
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
4.0 s. The second and third columns give the
value of t
1
= 
 
t
t
2
-
?
?
?
?
?
?
?
 and 
t t
t
2
2
= +
?
?
?
?
?
?
?
 and the
fourth and the fifth columns give the
2024-25
MOTION IN A STRAIGHT LINE 15
?
corresponding values of x, i.e. x (t
1
) = 0.08 t
1
3
and x (t
2
) = 0.08 
t
2
3
. The sixth column lists the
difference ?x = x (t
2
) – x (t
1
) and the last
column gives the ratio of ?x and ?t, i.e. the
average velocity corresponding to the value
of ?t listed in the first column.
We see from Table 2.1 that as we decrease
the value of ?t from 2.0 s to 0.010 s, the value of
the average velocity approaches the limiting
value 3.84 m s
–1
 which is the value of velocity at
t = 4.0 s, i.e. the value of  
dx
dt
 at t = 4.0 s. In this
manner, we can calculate velocity at each
instant for motion of the car.
The graphical method for the determination
of the instantaneous velocity is always not a
convenient method.  For this, we must carefully
plot the position–time graph and calculate the
value of average velocity as ?t becomes smaller
and smaller.  It is easier to calculate the value
of velocity at different instants if we have data
of positions at different instants or exact
expression for the position as a function of time.
Then, we calculate ?x/?t from the data for
decreasing the value of ?t and find the limiting
value as we have done in Table 2.1 or use
differential calculus for the given expression and
calculate 
dx
dt
 at different instants as done in
the following example.
Example 2.1    The position of an object
moving along x-axis is given by   x = a + bt
2
where  a = 8.5 m, b = 2.5 m s
–2
 and t is
measured in seconds. What is its velocity at
t = 0 s and t = 2.0 s. What is the average
velocity between t = 2.0 s and t = 4.0 s ?
Answer  In notation of differential calculus, the
velocity is
( ) 
2 -1
dx d
v a bt 2b t = 5.0 t m s
dt dt
= = + =
At   t = 0 s,      v = 0 m s
–1
   and at   t = 2.0 s,
v = 10 m s
-1
 .
( ) ( ) 4.0 2.0
4.0 2.0
x x
Average velocity
-
=
-
16 – – 4
6.0
2.0
a b a b
b
+
= = ×
      
-1
6.0 2.5 =15 m s = ×
            ?
Note that for uniform motion, velocity is
the same as the average velocity at all
instants.
Instantaneous speed or simply speed is the
magnitude of velocity. For example, a velocity of
+ 24.0 m s
–1
 and a velocity of – 24.0 m s
–1
 —
both have an associated speed of 24.0 m s
-1
.  It
should be noted that though average speed over
a finite interval of time is greater or equal to the
magnitude of the average velocity,
instantaneous speed at an instant is equal to
the magnitude of the instantaneous velocity at
that instant. Why so ?
2.3  ACCELERATION
The velocity of an object, in general, changes
during its course of motion. How to describe
this change? Should it be described as the rate
of change in velocity with distance or with
time ?   This was a problem even in Galileo’s
time. It was first thought that this change could
be described by the rate of change of velocity
with distance. But, through his studies of
motion of freely falling objects and motion of
objects on an inclined plane, Galileo concluded
that the rate of change of velocity with time is
a constant of motion for all objects in free fall.
On the other hand, the change in velocity with
distance is not constant – it decreases with the
increasing distance of fall. This led to the
concept of acceleration as the rate of change
of velocity with time.
 The average acceleration a over a time interval
is defined as the change of velocity divided by
the time interval :
2 1
2 1
–
–
v v v
a
t t t
?
= =
?
(2.2)
where v
2
 and v
1
 are the instantaneous velocities
or simply velocities at time  t
2
 
and t
1
 
. It is the
average change of velocity per unit time. The SI
unit of acceleration is m s
–2
 .
On a plot of velocity versus time, the average
acceleration is the slope of the straight line
connecting the points corresponding to (v
2
, t
2
)
and (v
1
, t
1
).
2024-25
Page 4


CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 Introduction
2.2 Instantaneous velocity and
speed
2.3 Acceleration
2.4 Kinematic equations for
uniformly accelerated motion
2.5 Relative velocity
Summary
Points to ponder
Exercises
2.1  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 4.
2024-25
PHYSICS 14
Table 2.1  Limiting value of 
?
?
x
t
 at t = 4 s
2.2  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,
v lim
x
t
=
t  0 ? 
?
? ?
(2.1a)
  
=
d
d
x
t
(2.1b)
where the symbol 
lim
t 0 ? ?
 stands for the operation
of taking limit as  ?tg0 of the quantity on its
right. In the language of calculus, the quantity
on the right hand side of Eq. (2.1a) is the
differential coefficient of x with respect to t and
is denoted by 
 
d
d
x
t
 (see Appendix 2.1).  It is the
rate of change of position with respect to time,
at that instant.
We can use Eq. (2.1a) 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.2.1 calculation.
Let us take ?t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the
slope of line P
1
P
2
  ( Fig. 2.1) gives the value of
average velocity over the interval   3 s to 5 s.
Fig. 2.1 Determining velocity from position-time
graph.  Velocity at t = 4 s is the slope of the
tangent to the graph at that instant.
Now, we decrease the value of ?t from 2 s to 1
s.  Then line P
1
P
2
 becomes Q
1
Q
2
   and its slope
gives the value of the average velocity over
the interval 3.5 s to 4.5 s. In the limit ?t ? 0,
the line P
1
P
2
 becomes tangent to the position-
time curve at the point P and the velocity at t
= 4 s is given by the slope of the tangent at
that point. It is difficult to show this
process graphically. But if we use
numerical method to obtain the value of
the velocity, the meaning of the limiting
process becomes clear. For the graph shown
in Fig. 2.1, x = 0.08 t
3
.  Table 2.1 gives the
value of ?x/?t calculated for ?t equal to 2.0 s,
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
4.0 s. The second and third columns give the
value of t
1
= 
 
t
t
2
-
?
?
?
?
?
?
?
 and 
t t
t
2
2
= +
?
?
?
?
?
?
?
 and the
fourth and the fifth columns give the
2024-25
MOTION IN A STRAIGHT LINE 15
?
corresponding values of x, i.e. x (t
1
) = 0.08 t
1
3
and x (t
2
) = 0.08 
t
2
3
. The sixth column lists the
difference ?x = x (t
2
) – x (t
1
) and the last
column gives the ratio of ?x and ?t, i.e. the
average velocity corresponding to the value
of ?t listed in the first column.
We see from Table 2.1 that as we decrease
the value of ?t from 2.0 s to 0.010 s, the value of
the average velocity approaches the limiting
value 3.84 m s
–1
 which is the value of velocity at
t = 4.0 s, i.e. the value of  
dx
dt
 at t = 4.0 s. In this
manner, we can calculate velocity at each
instant for motion of the car.
The graphical method for the determination
of the instantaneous velocity is always not a
convenient method.  For this, we must carefully
plot the position–time graph and calculate the
value of average velocity as ?t becomes smaller
and smaller.  It is easier to calculate the value
of velocity at different instants if we have data
of positions at different instants or exact
expression for the position as a function of time.
Then, we calculate ?x/?t from the data for
decreasing the value of ?t and find the limiting
value as we have done in Table 2.1 or use
differential calculus for the given expression and
calculate 
dx
dt
 at different instants as done in
the following example.
Example 2.1    The position of an object
moving along x-axis is given by   x = a + bt
2
where  a = 8.5 m, b = 2.5 m s
–2
 and t is
measured in seconds. What is its velocity at
t = 0 s and t = 2.0 s. What is the average
velocity between t = 2.0 s and t = 4.0 s ?
Answer  In notation of differential calculus, the
velocity is
( ) 
2 -1
dx d
v a bt 2b t = 5.0 t m s
dt dt
= = + =
At   t = 0 s,      v = 0 m s
–1
   and at   t = 2.0 s,
v = 10 m s
-1
 .
( ) ( ) 4.0 2.0
4.0 2.0
x x
Average velocity
-
=
-
16 – – 4
6.0
2.0
a b a b
b
+
= = ×
      
-1
6.0 2.5 =15 m s = ×
            ?
Note that for uniform motion, velocity is
the same as the average velocity at all
instants.
Instantaneous speed or simply speed is the
magnitude of velocity. For example, a velocity of
+ 24.0 m s
–1
 and a velocity of – 24.0 m s
–1
 —
both have an associated speed of 24.0 m s
-1
.  It
should be noted that though average speed over
a finite interval of time is greater or equal to the
magnitude of the average velocity,
instantaneous speed at an instant is equal to
the magnitude of the instantaneous velocity at
that instant. Why so ?
2.3  ACCELERATION
The velocity of an object, in general, changes
during its course of motion. How to describe
this change? Should it be described as the rate
of change in velocity with distance or with
time ?   This was a problem even in Galileo’s
time. It was first thought that this change could
be described by the rate of change of velocity
with distance. But, through his studies of
motion of freely falling objects and motion of
objects on an inclined plane, Galileo concluded
that the rate of change of velocity with time is
a constant of motion for all objects in free fall.
On the other hand, the change in velocity with
distance is not constant – it decreases with the
increasing distance of fall. This led to the
concept of acceleration as the rate of change
of velocity with time.
 The average acceleration a over a time interval
is defined as the change of velocity divided by
the time interval :
2 1
2 1
–
–
v v v
a
t t t
?
= =
?
(2.2)
where v
2
 and v
1
 are the instantaneous velocities
or simply velocities at time  t
2
 
and t
1
 
. It is the
average change of velocity per unit time. The SI
unit of acceleration is m s
–2
 .
On a plot of velocity versus time, the average
acceleration is the slope of the straight line
connecting the points corresponding to (v
2
, t
2
)
and (v
1
, t
1
).
2024-25
PHYSICS 16
Fig. 2.3 Velocity–time graph for motions with
constant acceleration. (a) Motion in positive
direction with positive acceleration,
(b) Motion in positive direction with
negative acceleration, (c) Motion in
negative direction with negative
acceleration, (d) Motion of an object with
negative acceleration that changes
direction at time t
1
.  Between times 0 to
t
1
, it moves in  positive x - direction
and between t
1
 and t
2
 it moves in the
opposite direction.
Instantaneous acceleration is defined in the
same way as the instantaneous velocity :
d
d
t 0
v v
a lim
t t
? ?
?
= =
?
(2.3)
The acceleration at an instant is the slope
of the tangent to the v–t curve at that
instant.
Since velocity is a quantity having both
magnitude and direction, a change in
velocity may involve either or both of these
factors.  Acceleration, therefore, may result
from a change in speed (magnitude), a
change in direction or changes in both.  Like
velocity, acceleration can also be positive,
negative or zero.  Position-time graphs for
motion with positive, negative and zero
acceleration are shown in Figs. 2.4 (a), (b)
and (c), respectively.  Note that the graph
curves upward for positive acceleration;
downward for negative acceleration and it is
a straight line for zero acceleration.
Although acceleration can vary with time,
our study in this chapter will be restricted
to motion with constant acceleration. In this
case, the average acceleration equals the
constant value of acceleration during the
interval. If the velocity of an object is v
o
 at t
= 0 and v at time t, we have
or
0
0
0
v v
a  ,  v v a t
t
- = = +
-        (2.4)
Fig. 2.2 Position-time graph for motion with
(a) positive acceleration; (b) negative
acceleration, and (c) zero acceleration.
Let us see how velocity-time graph looks like
for some simple cases. Fig. 2.3 shows     velocity-
time graph for motion with constant acceleration
for the following cases :
(a) An object is moving in a positive direction
with a positive acceleration.
(b) An object is moving in positive direction
with a negative acceleration.
(c) An object is moving in negative direction
with a negative acceleration.
(d) An object is moving in positive direction
till time t
1
, and then turns back with the
same negative acceleration.
An interesting feature of a velocity-time
graph for any moving object is that the area
under the curve represents the
displacement over a given time interval. A
general proof of this statement requires use of
calculus. We can, however, see that it is true
for the simple case of an object moving with
constant velocity u. Its velocity-time graph is
as shown in Fig. 2.4.
2024-25
Page 5


CHAPTER TWO
MOTION IN A STRAIGHT LINE
2.1 Introduction
2.2 Instantaneous velocity and
speed
2.3 Acceleration
2.4 Kinematic equations for
uniformly accelerated motion
2.5 Relative velocity
Summary
Points to ponder
Exercises
2.1  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 4.
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PHYSICS 14
Table 2.1  Limiting value of 
?
?
x
t
 at t = 4 s
2.2  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,
v lim
x
t
=
t  0 ? 
?
? ?
(2.1a)
  
=
d
d
x
t
(2.1b)
where the symbol 
lim
t 0 ? ?
 stands for the operation
of taking limit as  ?tg0 of the quantity on its
right. In the language of calculus, the quantity
on the right hand side of Eq. (2.1a) is the
differential coefficient of x with respect to t and
is denoted by 
 
d
d
x
t
 (see Appendix 2.1).  It is the
rate of change of position with respect to time,
at that instant.
We can use Eq. (2.1a) 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.2.1 calculation.
Let us take ?t = 2 s centred at t = 4 s. Then,
by the definition of the average velocity, the
slope of line P
1
P
2
  ( Fig. 2.1) gives the value of
average velocity over the interval   3 s to 5 s.
Fig. 2.1 Determining velocity from position-time
graph.  Velocity at t = 4 s is the slope of the
tangent to the graph at that instant.
Now, we decrease the value of ?t from 2 s to 1
s.  Then line P
1
P
2
 becomes Q
1
Q
2
   and its slope
gives the value of the average velocity over
the interval 3.5 s to 4.5 s. In the limit ?t ? 0,
the line P
1
P
2
 becomes tangent to the position-
time curve at the point P and the velocity at t
= 4 s is given by the slope of the tangent at
that point. It is difficult to show this
process graphically. But if we use
numerical method to obtain the value of
the velocity, the meaning of the limiting
process becomes clear. For the graph shown
in Fig. 2.1, x = 0.08 t
3
.  Table 2.1 gives the
value of ?x/?t calculated for ?t equal to 2.0 s,
1.0 s, 0.5 s, 0.1 s and 0.01 s centred at t =
4.0 s. The second and third columns give the
value of t
1
= 
 
t
t
2
-
?
?
?
?
?
?
?
 and 
t t
t
2
2
= +
?
?
?
?
?
?
?
 and the
fourth and the fifth columns give the
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MOTION IN A STRAIGHT LINE 15
?
corresponding values of x, i.e. x (t
1
) = 0.08 t
1
3
and x (t
2
) = 0.08 
t
2
3
. The sixth column lists the
difference ?x = x (t
2
) – x (t
1
) and the last
column gives the ratio of ?x and ?t, i.e. the
average velocity corresponding to the value
of ?t listed in the first column.
We see from Table 2.1 that as we decrease
the value of ?t from 2.0 s to 0.010 s, the value of
the average velocity approaches the limiting
value 3.84 m s
–1
 which is the value of velocity at
t = 4.0 s, i.e. the value of  
dx
dt
 at t = 4.0 s. In this
manner, we can calculate velocity at each
instant for motion of the car.
The graphical method for the determination
of the instantaneous velocity is always not a
convenient method.  For this, we must carefully
plot the position–time graph and calculate the
value of average velocity as ?t becomes smaller
and smaller.  It is easier to calculate the value
of velocity at different instants if we have data
of positions at different instants or exact
expression for the position as a function of time.
Then, we calculate ?x/?t from the data for
decreasing the value of ?t and find the limiting
value as we have done in Table 2.1 or use
differential calculus for the given expression and
calculate 
dx
dt
 at different instants as done in
the following example.
Example 2.1    The position of an object
moving along x-axis is given by   x = a + bt
2
where  a = 8.5 m, b = 2.5 m s
–2
 and t is
measured in seconds. What is its velocity at
t = 0 s and t = 2.0 s. What is the average
velocity between t = 2.0 s and t = 4.0 s ?
Answer  In notation of differential calculus, the
velocity is
( ) 
2 -1
dx d
v a bt 2b t = 5.0 t m s
dt dt
= = + =
At   t = 0 s,      v = 0 m s
–1
   and at   t = 2.0 s,
v = 10 m s
-1
 .
( ) ( ) 4.0 2.0
4.0 2.0
x x
Average velocity
-
=
-
16 – – 4
6.0
2.0
a b a b
b
+
= = ×
      
-1
6.0 2.5 =15 m s = ×
            ?
Note that for uniform motion, velocity is
the same as the average velocity at all
instants.
Instantaneous speed or simply speed is the
magnitude of velocity. For example, a velocity of
+ 24.0 m s
–1
 and a velocity of – 24.0 m s
–1
 —
both have an associated speed of 24.0 m s
-1
.  It
should be noted that though average speed over
a finite interval of time is greater or equal to the
magnitude of the average velocity,
instantaneous speed at an instant is equal to
the magnitude of the instantaneous velocity at
that instant. Why so ?
2.3  ACCELERATION
The velocity of an object, in general, changes
during its course of motion. How to describe
this change? Should it be described as the rate
of change in velocity with distance or with
time ?   This was a problem even in Galileo’s
time. It was first thought that this change could
be described by the rate of change of velocity
with distance. But, through his studies of
motion of freely falling objects and motion of
objects on an inclined plane, Galileo concluded
that the rate of change of velocity with time is
a constant of motion for all objects in free fall.
On the other hand, the change in velocity with
distance is not constant – it decreases with the
increasing distance of fall. This led to the
concept of acceleration as the rate of change
of velocity with time.
 The average acceleration a over a time interval
is defined as the change of velocity divided by
the time interval :
2 1
2 1
–
–
v v v
a
t t t
?
= =
?
(2.2)
where v
2
 and v
1
 are the instantaneous velocities
or simply velocities at time  t
2
 
and t
1
 
. It is the
average change of velocity per unit time. The SI
unit of acceleration is m s
–2
 .
On a plot of velocity versus time, the average
acceleration is the slope of the straight line
connecting the points corresponding to (v
2
, t
2
)
and (v
1
, t
1
).
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PHYSICS 16
Fig. 2.3 Velocity–time graph for motions with
constant acceleration. (a) Motion in positive
direction with positive acceleration,
(b) Motion in positive direction with
negative acceleration, (c) Motion in
negative direction with negative
acceleration, (d) Motion of an object with
negative acceleration that changes
direction at time t
1
.  Between times 0 to
t
1
, it moves in  positive x - direction
and between t
1
 and t
2
 it moves in the
opposite direction.
Instantaneous acceleration is defined in the
same way as the instantaneous velocity :
d
d
t 0
v v
a lim
t t
? ?
?
= =
?
(2.3)
The acceleration at an instant is the slope
of the tangent to the v–t curve at that
instant.
Since velocity is a quantity having both
magnitude and direction, a change in
velocity may involve either or both of these
factors.  Acceleration, therefore, may result
from a change in speed (magnitude), a
change in direction or changes in both.  Like
velocity, acceleration can also be positive,
negative or zero.  Position-time graphs for
motion with positive, negative and zero
acceleration are shown in Figs. 2.4 (a), (b)
and (c), respectively.  Note that the graph
curves upward for positive acceleration;
downward for negative acceleration and it is
a straight line for zero acceleration.
Although acceleration can vary with time,
our study in this chapter will be restricted
to motion with constant acceleration. In this
case, the average acceleration equals the
constant value of acceleration during the
interval. If the velocity of an object is v
o
 at t
= 0 and v at time t, we have
or
0
0
0
v v
a  ,  v v a t
t
- = = +
-        (2.4)
Fig. 2.2 Position-time graph for motion with
(a) positive acceleration; (b) negative
acceleration, and (c) zero acceleration.
Let us see how velocity-time graph looks like
for some simple cases. Fig. 2.3 shows     velocity-
time graph for motion with constant acceleration
for the following cases :
(a) An object is moving in a positive direction
with a positive acceleration.
(b) An object is moving in positive direction
with a negative acceleration.
(c) An object is moving in negative direction
with a negative acceleration.
(d) An object is moving in positive direction
till time t
1
, and then turns back with the
same negative acceleration.
An interesting feature of a velocity-time
graph for any moving object is that the area
under the curve represents the
displacement over a given time interval. A
general proof of this statement requires use of
calculus. We can, however, see that it is true
for the simple case of an object moving with
constant velocity u. Its velocity-time graph is
as shown in Fig. 2.4.
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MOTION IN A STRAIGHT LINE 17
Fig. 2.4 Area under v–t curve equals displacement
of the object over a given time interval.
The v-t curve is
 
a straight line parallel to the
time axis and the area under it between t = 0
and t = T
  
is the area of the rectangle of height u
and base T. Therefore, area = u × T = uT which
is the displacement in this time interval.  How
come in this case an area is equal to a distance?
Think!  Note the dimensions of quantities on
the two coordinate axes, and you will arrive at
the answer.
Note that the x-t, v-t, and a-t graphs shown
in several figures in this chapter have sharp
kinks at some points implying that the
functions are not differentiable at these
points. In any realistic situation, the
functions will be differentiable at all points
and the graphs will be smooth.
What this means physically is that
acceleration and velocity cannot change
values abruptly at an instant. Changes are
always continuous.
2.4 KINEMATIC EQUATIONS FOR
UNIFORMLY ACCELERATED MOTION
For uniformly accelerated motion, we can derive
some simple equations that relate displacement
(x), time taken (t), initial velocity (v
0
), final
velocity (v) and acceleration (a). Equation (2.4)
already obtained gives a relation between final
and initial velocities v and  v
0  
of an object moving
with uniform acceleration
 
a :
             v = v
0
 + at (2.4)
This relation is graphically represented in Fig. 2.5.
The area under this curve is :
Area between instants 0 and t = Area of triangle
ABC + Area of rectangle OACD
 
( ) –
0 0
1
v v t + v t
2
=
Fig. 2.5 Area  under v-t curve for an object with
uniform acceleration.
As explained in the previous section, the area
under v-t curve represents the displacement.
Therefore, the displacement x of the object is :
( )
1
–
2
0 0
x v v t + v t = (2.5)
But
0
v v a t - =
Therefore,
2
0
1
2
x a t + v t =
or,
2
0
1
2
x v t at = +
(2.6)
Equation (2.5) can also be written as
0
2
v + v
x t v t = = (2.7a)
where,
0
2
v v
v
+
=
  (constant acceleration only)
(2.7b)
Equations (2.7a) and  (2.7b) mean that the object
has undergone displacement x with an average
velocity equal to the arithmetic average of the
initial and final velocities.
From Eq. (2.4), t = (v – v
0
)/a. Substituting this in
Eq. (2.7a), we get
       
x v t
v v v v
a
v v
a
= =
+ ?
?
?
?
?
?
- ?
?
?
?
?
?
=
-
0 0
2
0
2
2 2
     
2 2
0
2 v v ax = + (2.8)
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