Page 1
CHAPTER FOUR
MOTION IN A PLANE
4.1 INTRODUCTION
In the last chapter we developed the concepts of position,
displacement, velocity and acceleration that are needed to
describe the motion of an object along a straight line. We
found that the directional aspect of these quantities can be
taken care of by + and – signs, as in one dimension only two
directions are possible. But in order to describe motion of an
object in two dimensions (a plane) or three dimensions
(space), we need to use vectors to describe the above
mentioned physical quantities. Therefore, it is first necessary
to learn the language of vectors. What is a vector? How to
add, subtract and multiply vectors ? What is the result of
multiplying a vector by a real number ? We shall learn this
to enable us to use vectors for defining velocity and
acceleration in a plane. We then discuss motion of an object
in a plane. As a simple case of motion in a plane, we shall
discuss motion with constant acceleration and treat in detail
the projectile motion. Circular motion is a familiar class of
motion that has a special significance in dailylife situations.
We shall discuss uniform circular motion in some detail.
The equations developed in this chapter for motion in a
plane can be easily extended to the case of three dimensions.
4.2 SCALARS AND VECTORS
In physics, we can classify quantities as scalars or
vectors. Basically, the difference is that a direction is
associated with a vector but not with a scalar. A scalar
quantity is a quantity with magnitude only. It is specified
completely by a single number, along with the pr oper
unit. Examples are : the distance between two points,
mass of an object, the temperature of a body and the
time at which a certain event happened. The rules for
combining scalars are the rules of ordinary algebra.
Scalars can be added, subtracted, multiplied and divided
4.1 Introduction
4.2 Scalars and vectors
4.3 Multiplication of vectors by
real numbers
4.4 Addition and subtraction of
vectors — graphical method
4.5 Resolution of vectors
4.6 Vector addition — analytical
method
4.7 Motion in a plane
4.8 Motion in a plane with
constant acceleration
4.9 Relative velocity in two
dimensions
4.10 Projectile motion
4.11 Uniform circular motion
Summary
Points to ponder
Exercises
Additional exercises
202223
Page 2
CHAPTER FOUR
MOTION IN A PLANE
4.1 INTRODUCTION
In the last chapter we developed the concepts of position,
displacement, velocity and acceleration that are needed to
describe the motion of an object along a straight line. We
found that the directional aspect of these quantities can be
taken care of by + and – signs, as in one dimension only two
directions are possible. But in order to describe motion of an
object in two dimensions (a plane) or three dimensions
(space), we need to use vectors to describe the above
mentioned physical quantities. Therefore, it is first necessary
to learn the language of vectors. What is a vector? How to
add, subtract and multiply vectors ? What is the result of
multiplying a vector by a real number ? We shall learn this
to enable us to use vectors for defining velocity and
acceleration in a plane. We then discuss motion of an object
in a plane. As a simple case of motion in a plane, we shall
discuss motion with constant acceleration and treat in detail
the projectile motion. Circular motion is a familiar class of
motion that has a special significance in dailylife situations.
We shall discuss uniform circular motion in some detail.
The equations developed in this chapter for motion in a
plane can be easily extended to the case of three dimensions.
4.2 SCALARS AND VECTORS
In physics, we can classify quantities as scalars or
vectors. Basically, the difference is that a direction is
associated with a vector but not with a scalar. A scalar
quantity is a quantity with magnitude only. It is specified
completely by a single number, along with the pr oper
unit. Examples are : the distance between two points,
mass of an object, the temperature of a body and the
time at which a certain event happened. The rules for
combining scalars are the rules of ordinary algebra.
Scalars can be added, subtracted, multiplied and divided
4.1 Introduction
4.2 Scalars and vectors
4.3 Multiplication of vectors by
real numbers
4.4 Addition and subtraction of
vectors — graphical method
4.5 Resolution of vectors
4.6 Vector addition — analytical
method
4.7 Motion in a plane
4.8 Motion in a plane with
constant acceleration
4.9 Relative velocity in two
dimensions
4.10 Projectile motion
4.11 Uniform circular motion
Summary
Points to ponder
Exercises
Additional exercises
202223
PHYSICS 66
just as the ordinary numbers*. For example,
if the length and breadth of a rectangle are
1.0 m and 0.5 m respectively, then its
perimeter is the sum of the lengths of the
four sides, 1.0 m + 0.5 m +1.0 m + 0.5 m =
3.0 m. The length of each side is a scalar
and the perimeter is also a scalar. Take
another example: the maximum and
minimum temperatures on a particular day
are 35.6 °C and 24.2 °C respectively. Then,
the difference between the two temperatures
is 11.4 °C. Similarly, if a uniform solid cube
of aluminium of side 10 cm has a mass of
2.7 kg, then its volume is 10
–3
m
3
(a scalar)
and its density is 2.7 ×10
3
kg m
–3
(a scalar).
A vector quantity is a quantity that has both
a magnitude and a direction and obeys the
triangle law of addition or equivalently the
parallelogram law of addition. So, a vector is
specified by giving its magnitude by a number
and its direction. Some physical quantities that
are represented by vectors are displacement,
velocity, acceleration and force.
To represent a vector, we use a bold face type
in this book. Thus, a velocity vector can be
represented by a symbol v. Since bold face is
difficult to produce, when written by hand, a
vector is often represented by an arrow placed
over a letter, say
r
v
. Thus, both v and
r
v
represent the velocity vector. The magnitude of
a vector is often called its absolute value,
indicated by  v = v. Thus, a vector is
represented by a bold face, e.g. by A, a, p, q, r, ...
x, y, with respective magnitudes denoted by light
face A, a, p, q, r, ... x, y.
4.2.1 Position and Displacement Vectors
To describe the position of an object moving in
a plane, we need to choose a convenient point,
say O as origin. Let P and P' be the positions of
the object at time t and t', respectively [Fig. 4.1(a)].
We join O and P by a straight line. Then, OP is
the position vector of the object at time t. An
arrow is marked at the head of this line. It is
represented by a symbol r, i.e. OP = r. Point P' is
represented by another position vector, OP'
denoted by r'. The length of the vector r
represents the magnitude of the vector and its
direction is the direction in which P lies as seen
from O. If the object moves from P to P', the
vector PP' (with tail at P and tip at P') is called
the displacement vector corresponding to
motion from point P (at time t) to point P' (at time t').
Fig. 4.1 (a) Position and displacement vectors.
(b) Displacement vector PQ and different
courses of motion.
It is important to note that displacement
vector is the straight line joining the initial and
final positions and does not depend on the actual
path undertaken by the object between the two
positions. For example, in Fig. 4.1(b), given the
initial and final positions as P and Q, the
displacement vector is the same PQ for different
paths of journey, say PABCQ, PDQ, and PBEFQ.
Therefore, the magnitude of displacement is
either less or equal to the path length of an
object between two points. This fact was
emphasised in the previous chapter also while
discussing motion along a straight line.
4.2.2 Equality of Vectors
Two vectors A and B are said to be equal if, and
only if, they have the same magnitude and the
same direction.**
Figure 4.2(a) shows two equal vectors A and
B. We can easily check their equality. Shift B
parallel to itself until its tail Q coincides with that
of A, i.e. Q coincides with O. Then, since their
tips S and P also coincide, the two vectors are
said to be equal. In general, equality is indicated
* Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply
and divide scalars of different units.
** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector
unchanged. Such vectors are called free vectors. However, in some physical applications, location or line of
application of a vector is important. Such vectors are called localised vectors.
202223
Page 3
CHAPTER FOUR
MOTION IN A PLANE
4.1 INTRODUCTION
In the last chapter we developed the concepts of position,
displacement, velocity and acceleration that are needed to
describe the motion of an object along a straight line. We
found that the directional aspect of these quantities can be
taken care of by + and – signs, as in one dimension only two
directions are possible. But in order to describe motion of an
object in two dimensions (a plane) or three dimensions
(space), we need to use vectors to describe the above
mentioned physical quantities. Therefore, it is first necessary
to learn the language of vectors. What is a vector? How to
add, subtract and multiply vectors ? What is the result of
multiplying a vector by a real number ? We shall learn this
to enable us to use vectors for defining velocity and
acceleration in a plane. We then discuss motion of an object
in a plane. As a simple case of motion in a plane, we shall
discuss motion with constant acceleration and treat in detail
the projectile motion. Circular motion is a familiar class of
motion that has a special significance in dailylife situations.
We shall discuss uniform circular motion in some detail.
The equations developed in this chapter for motion in a
plane can be easily extended to the case of three dimensions.
4.2 SCALARS AND VECTORS
In physics, we can classify quantities as scalars or
vectors. Basically, the difference is that a direction is
associated with a vector but not with a scalar. A scalar
quantity is a quantity with magnitude only. It is specified
completely by a single number, along with the pr oper
unit. Examples are : the distance between two points,
mass of an object, the temperature of a body and the
time at which a certain event happened. The rules for
combining scalars are the rules of ordinary algebra.
Scalars can be added, subtracted, multiplied and divided
4.1 Introduction
4.2 Scalars and vectors
4.3 Multiplication of vectors by
real numbers
4.4 Addition and subtraction of
vectors — graphical method
4.5 Resolution of vectors
4.6 Vector addition — analytical
method
4.7 Motion in a plane
4.8 Motion in a plane with
constant acceleration
4.9 Relative velocity in two
dimensions
4.10 Projectile motion
4.11 Uniform circular motion
Summary
Points to ponder
Exercises
Additional exercises
202223
PHYSICS 66
just as the ordinary numbers*. For example,
if the length and breadth of a rectangle are
1.0 m and 0.5 m respectively, then its
perimeter is the sum of the lengths of the
four sides, 1.0 m + 0.5 m +1.0 m + 0.5 m =
3.0 m. The length of each side is a scalar
and the perimeter is also a scalar. Take
another example: the maximum and
minimum temperatures on a particular day
are 35.6 °C and 24.2 °C respectively. Then,
the difference between the two temperatures
is 11.4 °C. Similarly, if a uniform solid cube
of aluminium of side 10 cm has a mass of
2.7 kg, then its volume is 10
–3
m
3
(a scalar)
and its density is 2.7 ×10
3
kg m
–3
(a scalar).
A vector quantity is a quantity that has both
a magnitude and a direction and obeys the
triangle law of addition or equivalently the
parallelogram law of addition. So, a vector is
specified by giving its magnitude by a number
and its direction. Some physical quantities that
are represented by vectors are displacement,
velocity, acceleration and force.
To represent a vector, we use a bold face type
in this book. Thus, a velocity vector can be
represented by a symbol v. Since bold face is
difficult to produce, when written by hand, a
vector is often represented by an arrow placed
over a letter, say
r
v
. Thus, both v and
r
v
represent the velocity vector. The magnitude of
a vector is often called its absolute value,
indicated by  v = v. Thus, a vector is
represented by a bold face, e.g. by A, a, p, q, r, ...
x, y, with respective magnitudes denoted by light
face A, a, p, q, r, ... x, y.
4.2.1 Position and Displacement Vectors
To describe the position of an object moving in
a plane, we need to choose a convenient point,
say O as origin. Let P and P' be the positions of
the object at time t and t', respectively [Fig. 4.1(a)].
We join O and P by a straight line. Then, OP is
the position vector of the object at time t. An
arrow is marked at the head of this line. It is
represented by a symbol r, i.e. OP = r. Point P' is
represented by another position vector, OP'
denoted by r'. The length of the vector r
represents the magnitude of the vector and its
direction is the direction in which P lies as seen
from O. If the object moves from P to P', the
vector PP' (with tail at P and tip at P') is called
the displacement vector corresponding to
motion from point P (at time t) to point P' (at time t').
Fig. 4.1 (a) Position and displacement vectors.
(b) Displacement vector PQ and different
courses of motion.
It is important to note that displacement
vector is the straight line joining the initial and
final positions and does not depend on the actual
path undertaken by the object between the two
positions. For example, in Fig. 4.1(b), given the
initial and final positions as P and Q, the
displacement vector is the same PQ for different
paths of journey, say PABCQ, PDQ, and PBEFQ.
Therefore, the magnitude of displacement is
either less or equal to the path length of an
object between two points. This fact was
emphasised in the previous chapter also while
discussing motion along a straight line.
4.2.2 Equality of Vectors
Two vectors A and B are said to be equal if, and
only if, they have the same magnitude and the
same direction.**
Figure 4.2(a) shows two equal vectors A and
B. We can easily check their equality. Shift B
parallel to itself until its tail Q coincides with that
of A, i.e. Q coincides with O. Then, since their
tips S and P also coincide, the two vectors are
said to be equal. In general, equality is indicated
* Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply
and divide scalars of different units.
** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector
unchanged. Such vectors are called free vectors. However, in some physical applications, location or line of
application of a vector is important. Such vectors are called localised vectors.
202223
MOTION IN A PLANE 67
as A = B. Note that in Fig. 4.2(b), vectors A' and
B' have the same magnitude but they are not
equal because they have different directions.
Even if we shift B' parallel to itself so that its tail
Q' coincides with the tail O ' of A' , the tip S' of B'
does not coincide with the tip P' of A' .
4.3 MULTIPLICATION OF VECTORS BY REAL
NUMBERS
Multiplying a vector A with a positive number ?
gives a vector whose magnitude is changed by
the factor ? but the direction is the same as that
of A :
? ? A? = ? ? A? if ? > 0.
For example, if A is multiplied by 2, the resultant
vector 2A is in the same direction as A and has
a magnitude twice of A as shown in Fig. 4.3(a).
Multiplying a vector A by a negative number
? gives another vector whose direction is
opposite to the direction of A and whose
magnitude is ? times A.
Multiplying a given vector A by negative
numbers, say –1 and –1.5, gives vectors as
shown in Fig 4.3(b).
The factor ? by which a vector A is multiplied
could be a scalar having its own physical
dimension. Then, the dimension of ? A is the
product of the dimensions of ? and A. For
example, if we multiply a constant velocity vector
by duration (of time), we get a displacement
vector.
4.4 ADDITION AND SUBTRACTION OF
VECTORS — GRAPHICAL METHOD
As mentioned in section 4.2, vectors, by
definition, obey the triangle law or equivalently,
the parallelogram law of addition. We shall now
describe this law of addition using the graphical
method. Let us consider two vectors A and B that
lie in a plane as shown in Fig. 4.4(a). The lengths
of the line segments representing these vectors
are proportional to the magnitude of the vectors.
To find the sum A + B, we place vector B so that
its tail is at the head of the vector A, as in
Fig. 4.4(b). Then, we join the tail of A to the head
of B. This line OQ represents a vector R, that is,
the sum of the vectors A and B. Since, in this
procedure of vector addition, vectors are
Fig. 4.2 (a) Two equal vectors A and B. (b) Two
vectors A' and B' are unequal though they
are of the same length.
Fig. 4.3 (a) Vector A and the resultant vector after
multiplying A by a positive number 2.
(b) Vector A and resultant vectors after
multiplying it by a negative number –1
and –1.5.
(c) (d)
Fig. 4.4 (a) Vectors A and B. (b) Vectors A and B
added graphically. (c) Vectors B and A
added graphically. (d) Illustrating the
associative law of vector addition.
202223
Page 4
CHAPTER FOUR
MOTION IN A PLANE
4.1 INTRODUCTION
In the last chapter we developed the concepts of position,
displacement, velocity and acceleration that are needed to
describe the motion of an object along a straight line. We
found that the directional aspect of these quantities can be
taken care of by + and – signs, as in one dimension only two
directions are possible. But in order to describe motion of an
object in two dimensions (a plane) or three dimensions
(space), we need to use vectors to describe the above
mentioned physical quantities. Therefore, it is first necessary
to learn the language of vectors. What is a vector? How to
add, subtract and multiply vectors ? What is the result of
multiplying a vector by a real number ? We shall learn this
to enable us to use vectors for defining velocity and
acceleration in a plane. We then discuss motion of an object
in a plane. As a simple case of motion in a plane, we shall
discuss motion with constant acceleration and treat in detail
the projectile motion. Circular motion is a familiar class of
motion that has a special significance in dailylife situations.
We shall discuss uniform circular motion in some detail.
The equations developed in this chapter for motion in a
plane can be easily extended to the case of three dimensions.
4.2 SCALARS AND VECTORS
In physics, we can classify quantities as scalars or
vectors. Basically, the difference is that a direction is
associated with a vector but not with a scalar. A scalar
quantity is a quantity with magnitude only. It is specified
completely by a single number, along with the pr oper
unit. Examples are : the distance between two points,
mass of an object, the temperature of a body and the
time at which a certain event happened. The rules for
combining scalars are the rules of ordinary algebra.
Scalars can be added, subtracted, multiplied and divided
4.1 Introduction
4.2 Scalars and vectors
4.3 Multiplication of vectors by
real numbers
4.4 Addition and subtraction of
vectors — graphical method
4.5 Resolution of vectors
4.6 Vector addition — analytical
method
4.7 Motion in a plane
4.8 Motion in a plane with
constant acceleration
4.9 Relative velocity in two
dimensions
4.10 Projectile motion
4.11 Uniform circular motion
Summary
Points to ponder
Exercises
Additional exercises
202223
PHYSICS 66
just as the ordinary numbers*. For example,
if the length and breadth of a rectangle are
1.0 m and 0.5 m respectively, then its
perimeter is the sum of the lengths of the
four sides, 1.0 m + 0.5 m +1.0 m + 0.5 m =
3.0 m. The length of each side is a scalar
and the perimeter is also a scalar. Take
another example: the maximum and
minimum temperatures on a particular day
are 35.6 °C and 24.2 °C respectively. Then,
the difference between the two temperatures
is 11.4 °C. Similarly, if a uniform solid cube
of aluminium of side 10 cm has a mass of
2.7 kg, then its volume is 10
–3
m
3
(a scalar)
and its density is 2.7 ×10
3
kg m
–3
(a scalar).
A vector quantity is a quantity that has both
a magnitude and a direction and obeys the
triangle law of addition or equivalently the
parallelogram law of addition. So, a vector is
specified by giving its magnitude by a number
and its direction. Some physical quantities that
are represented by vectors are displacement,
velocity, acceleration and force.
To represent a vector, we use a bold face type
in this book. Thus, a velocity vector can be
represented by a symbol v. Since bold face is
difficult to produce, when written by hand, a
vector is often represented by an arrow placed
over a letter, say
r
v
. Thus, both v and
r
v
represent the velocity vector. The magnitude of
a vector is often called its absolute value,
indicated by  v = v. Thus, a vector is
represented by a bold face, e.g. by A, a, p, q, r, ...
x, y, with respective magnitudes denoted by light
face A, a, p, q, r, ... x, y.
4.2.1 Position and Displacement Vectors
To describe the position of an object moving in
a plane, we need to choose a convenient point,
say O as origin. Let P and P' be the positions of
the object at time t and t', respectively [Fig. 4.1(a)].
We join O and P by a straight line. Then, OP is
the position vector of the object at time t. An
arrow is marked at the head of this line. It is
represented by a symbol r, i.e. OP = r. Point P' is
represented by another position vector, OP'
denoted by r'. The length of the vector r
represents the magnitude of the vector and its
direction is the direction in which P lies as seen
from O. If the object moves from P to P', the
vector PP' (with tail at P and tip at P') is called
the displacement vector corresponding to
motion from point P (at time t) to point P' (at time t').
Fig. 4.1 (a) Position and displacement vectors.
(b) Displacement vector PQ and different
courses of motion.
It is important to note that displacement
vector is the straight line joining the initial and
final positions and does not depend on the actual
path undertaken by the object between the two
positions. For example, in Fig. 4.1(b), given the
initial and final positions as P and Q, the
displacement vector is the same PQ for different
paths of journey, say PABCQ, PDQ, and PBEFQ.
Therefore, the magnitude of displacement is
either less or equal to the path length of an
object between two points. This fact was
emphasised in the previous chapter also while
discussing motion along a straight line.
4.2.2 Equality of Vectors
Two vectors A and B are said to be equal if, and
only if, they have the same magnitude and the
same direction.**
Figure 4.2(a) shows two equal vectors A and
B. We can easily check their equality. Shift B
parallel to itself until its tail Q coincides with that
of A, i.e. Q coincides with O. Then, since their
tips S and P also coincide, the two vectors are
said to be equal. In general, equality is indicated
* Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply
and divide scalars of different units.
** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector
unchanged. Such vectors are called free vectors. However, in some physical applications, location or line of
application of a vector is important. Such vectors are called localised vectors.
202223
MOTION IN A PLANE 67
as A = B. Note that in Fig. 4.2(b), vectors A' and
B' have the same magnitude but they are not
equal because they have different directions.
Even if we shift B' parallel to itself so that its tail
Q' coincides with the tail O ' of A' , the tip S' of B'
does not coincide with the tip P' of A' .
4.3 MULTIPLICATION OF VECTORS BY REAL
NUMBERS
Multiplying a vector A with a positive number ?
gives a vector whose magnitude is changed by
the factor ? but the direction is the same as that
of A :
? ? A? = ? ? A? if ? > 0.
For example, if A is multiplied by 2, the resultant
vector 2A is in the same direction as A and has
a magnitude twice of A as shown in Fig. 4.3(a).
Multiplying a vector A by a negative number
? gives another vector whose direction is
opposite to the direction of A and whose
magnitude is ? times A.
Multiplying a given vector A by negative
numbers, say –1 and –1.5, gives vectors as
shown in Fig 4.3(b).
The factor ? by which a vector A is multiplied
could be a scalar having its own physical
dimension. Then, the dimension of ? A is the
product of the dimensions of ? and A. For
example, if we multiply a constant velocity vector
by duration (of time), we get a displacement
vector.
4.4 ADDITION AND SUBTRACTION OF
VECTORS — GRAPHICAL METHOD
As mentioned in section 4.2, vectors, by
definition, obey the triangle law or equivalently,
the parallelogram law of addition. We shall now
describe this law of addition using the graphical
method. Let us consider two vectors A and B that
lie in a plane as shown in Fig. 4.4(a). The lengths
of the line segments representing these vectors
are proportional to the magnitude of the vectors.
To find the sum A + B, we place vector B so that
its tail is at the head of the vector A, as in
Fig. 4.4(b). Then, we join the tail of A to the head
of B. This line OQ represents a vector R, that is,
the sum of the vectors A and B. Since, in this
procedure of vector addition, vectors are
Fig. 4.2 (a) Two equal vectors A and B. (b) Two
vectors A' and B' are unequal though they
are of the same length.
Fig. 4.3 (a) Vector A and the resultant vector after
multiplying A by a positive number 2.
(b) Vector A and resultant vectors after
multiplying it by a negative number –1
and –1.5.
(c) (d)
Fig. 4.4 (a) Vectors A and B. (b) Vectors A and B
added graphically. (c) Vectors B and A
added graphically. (d) Illustrating the
associative law of vector addition.
202223
PHYSICS 68
arranged head to tail, this graphical method is
called the headtotail method. The two vectors
and their resultant form three sides of a triangle,
so this method is also known as triangle method
of vector addition. If we find the resultant of
B + A as in Fig. 4.4(c), the same vector R is
obtained. Thus, vector addition is commutative:
A + B = B + A (4.1)
The addition of vectors also obeys the associative
law as illustrated in Fig. 4.4(d). The result of
adding vectors A and B first and then adding
vector C is the same as the result of adding B
and C first and then adding vector A :
(A + B) + C = A + (B + C) (4.2)
What is the result of adding two equal and
opposite vectors ? Consider two vectors A and
–A shown in Fig. 4.3(b). Their sum is A + (–A).
Since the magnitudes of the two vectors are the
same, but the directions are opposite, the
resultant vector has zero magnitude and is
represented by 0 called a null vector or a zero
vector :
A – A = 0 0= 0 (4.3)
Since the magnitude of a null vector is zero, its
direction cannot be specified.
The null vector also results when we multiply
a vector A by the number zero. The main
properties of 0 are :
A + 0 = A
? 0 = 0
0 A = 0 (4.4)
Fig. 4.5 (a) Two vectors A and B, – B is also shown. (b) Subtracting vector B from vector A – the result is R
2
. For
comparison, addition of vectors A and B, i.e. R
1
is also shown.
What is the physical meaning of a zero vector?
Consider the position and displacement vectors
in a plane as shown in Fig. 4.1(a). Now suppose
that an object which is at P at time t, moves to
P' and then comes back to P. Then, what is its
displacement? Since the initial and final
positions coincide, the displacement is a “null
vector”.
Subtraction of vectors can be defined in terms
of addition of vectors. We define the difference
of two vectors A and B as the sum of two vectors
A and –B :
A – B = A + (–B) (4.5)
It is shown in Fig 4.5. The vector –B is added to
vector A to get R
2
= (A – B). The vector R
1
= A + B
is also shown in the same figure for comparison.
We can also use the parallelogram method to
find the sum of two vectors. Suppose we have
two vectors A and B. To add these vectors, we
bring their tails to a common origin O as
shown in Fig. 4.6(a). Then we draw a line from
the head of A parallel to B and another line from
the head of B parallel to A to complete a
parallelogram OQSP. Now we join the point of
the intersection of these two lines to the origin
O. The resultant vector R is directed from the
common origin O along the diagonal (OS) of the
parallelogram [ Fig. 4.6(b)]. In Fig.4.6(c), the
triangle law is used to obtain the resultant of A
and B and we see that the two methods yield the
same result. Thus, the two methods are
equivalent.
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CHAPTER FOUR
MOTION IN A PLANE
4.1 INTRODUCTION
In the last chapter we developed the concepts of position,
displacement, velocity and acceleration that are needed to
describe the motion of an object along a straight line. We
found that the directional aspect of these quantities can be
taken care of by + and – signs, as in one dimension only two
directions are possible. But in order to describe motion of an
object in two dimensions (a plane) or three dimensions
(space), we need to use vectors to describe the above
mentioned physical quantities. Therefore, it is first necessary
to learn the language of vectors. What is a vector? How to
add, subtract and multiply vectors ? What is the result of
multiplying a vector by a real number ? We shall learn this
to enable us to use vectors for defining velocity and
acceleration in a plane. We then discuss motion of an object
in a plane. As a simple case of motion in a plane, we shall
discuss motion with constant acceleration and treat in detail
the projectile motion. Circular motion is a familiar class of
motion that has a special significance in dailylife situations.
We shall discuss uniform circular motion in some detail.
The equations developed in this chapter for motion in a
plane can be easily extended to the case of three dimensions.
4.2 SCALARS AND VECTORS
In physics, we can classify quantities as scalars or
vectors. Basically, the difference is that a direction is
associated with a vector but not with a scalar. A scalar
quantity is a quantity with magnitude only. It is specified
completely by a single number, along with the pr oper
unit. Examples are : the distance between two points,
mass of an object, the temperature of a body and the
time at which a certain event happened. The rules for
combining scalars are the rules of ordinary algebra.
Scalars can be added, subtracted, multiplied and divided
4.1 Introduction
4.2 Scalars and vectors
4.3 Multiplication of vectors by
real numbers
4.4 Addition and subtraction of
vectors — graphical method
4.5 Resolution of vectors
4.6 Vector addition — analytical
method
4.7 Motion in a plane
4.8 Motion in a plane with
constant acceleration
4.9 Relative velocity in two
dimensions
4.10 Projectile motion
4.11 Uniform circular motion
Summary
Points to ponder
Exercises
Additional exercises
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PHYSICS 66
just as the ordinary numbers*. For example,
if the length and breadth of a rectangle are
1.0 m and 0.5 m respectively, then its
perimeter is the sum of the lengths of the
four sides, 1.0 m + 0.5 m +1.0 m + 0.5 m =
3.0 m. The length of each side is a scalar
and the perimeter is also a scalar. Take
another example: the maximum and
minimum temperatures on a particular day
are 35.6 °C and 24.2 °C respectively. Then,
the difference between the two temperatures
is 11.4 °C. Similarly, if a uniform solid cube
of aluminium of side 10 cm has a mass of
2.7 kg, then its volume is 10
–3
m
3
(a scalar)
and its density is 2.7 ×10
3
kg m
–3
(a scalar).
A vector quantity is a quantity that has both
a magnitude and a direction and obeys the
triangle law of addition or equivalently the
parallelogram law of addition. So, a vector is
specified by giving its magnitude by a number
and its direction. Some physical quantities that
are represented by vectors are displacement,
velocity, acceleration and force.
To represent a vector, we use a bold face type
in this book. Thus, a velocity vector can be
represented by a symbol v. Since bold face is
difficult to produce, when written by hand, a
vector is often represented by an arrow placed
over a letter, say
r
v
. Thus, both v and
r
v
represent the velocity vector. The magnitude of
a vector is often called its absolute value,
indicated by  v = v. Thus, a vector is
represented by a bold face, e.g. by A, a, p, q, r, ...
x, y, with respective magnitudes denoted by light
face A, a, p, q, r, ... x, y.
4.2.1 Position and Displacement Vectors
To describe the position of an object moving in
a plane, we need to choose a convenient point,
say O as origin. Let P and P' be the positions of
the object at time t and t', respectively [Fig. 4.1(a)].
We join O and P by a straight line. Then, OP is
the position vector of the object at time t. An
arrow is marked at the head of this line. It is
represented by a symbol r, i.e. OP = r. Point P' is
represented by another position vector, OP'
denoted by r'. The length of the vector r
represents the magnitude of the vector and its
direction is the direction in which P lies as seen
from O. If the object moves from P to P', the
vector PP' (with tail at P and tip at P') is called
the displacement vector corresponding to
motion from point P (at time t) to point P' (at time t').
Fig. 4.1 (a) Position and displacement vectors.
(b) Displacement vector PQ and different
courses of motion.
It is important to note that displacement
vector is the straight line joining the initial and
final positions and does not depend on the actual
path undertaken by the object between the two
positions. For example, in Fig. 4.1(b), given the
initial and final positions as P and Q, the
displacement vector is the same PQ for different
paths of journey, say PABCQ, PDQ, and PBEFQ.
Therefore, the magnitude of displacement is
either less or equal to the path length of an
object between two points. This fact was
emphasised in the previous chapter also while
discussing motion along a straight line.
4.2.2 Equality of Vectors
Two vectors A and B are said to be equal if, and
only if, they have the same magnitude and the
same direction.**
Figure 4.2(a) shows two equal vectors A and
B. We can easily check their equality. Shift B
parallel to itself until its tail Q coincides with that
of A, i.e. Q coincides with O. Then, since their
tips S and P also coincide, the two vectors are
said to be equal. In general, equality is indicated
* Addition and subtraction of scalars make sense only for quantities with same units. However, you can multiply
and divide scalars of different units.
** In our study, vectors do not have fixed locations. So displacing a vector parallel to itself leaves the vector
unchanged. Such vectors are called free vectors. However, in some physical applications, location or line of
application of a vector is important. Such vectors are called localised vectors.
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MOTION IN A PLANE 67
as A = B. Note that in Fig. 4.2(b), vectors A' and
B' have the same magnitude but they are not
equal because they have different directions.
Even if we shift B' parallel to itself so that its tail
Q' coincides with the tail O ' of A' , the tip S' of B'
does not coincide with the tip P' of A' .
4.3 MULTIPLICATION OF VECTORS BY REAL
NUMBERS
Multiplying a vector A with a positive number ?
gives a vector whose magnitude is changed by
the factor ? but the direction is the same as that
of A :
? ? A? = ? ? A? if ? > 0.
For example, if A is multiplied by 2, the resultant
vector 2A is in the same direction as A and has
a magnitude twice of A as shown in Fig. 4.3(a).
Multiplying a vector A by a negative number
? gives another vector whose direction is
opposite to the direction of A and whose
magnitude is ? times A.
Multiplying a given vector A by negative
numbers, say –1 and –1.5, gives vectors as
shown in Fig 4.3(b).
The factor ? by which a vector A is multiplied
could be a scalar having its own physical
dimension. Then, the dimension of ? A is the
product of the dimensions of ? and A. For
example, if we multiply a constant velocity vector
by duration (of time), we get a displacement
vector.
4.4 ADDITION AND SUBTRACTION OF
VECTORS — GRAPHICAL METHOD
As mentioned in section 4.2, vectors, by
definition, obey the triangle law or equivalently,
the parallelogram law of addition. We shall now
describe this law of addition using the graphical
method. Let us consider two vectors A and B that
lie in a plane as shown in Fig. 4.4(a). The lengths
of the line segments representing these vectors
are proportional to the magnitude of the vectors.
To find the sum A + B, we place vector B so that
its tail is at the head of the vector A, as in
Fig. 4.4(b). Then, we join the tail of A to the head
of B. This line OQ represents a vector R, that is,
the sum of the vectors A and B. Since, in this
procedure of vector addition, vectors are
Fig. 4.2 (a) Two equal vectors A and B. (b) Two
vectors A' and B' are unequal though they
are of the same length.
Fig. 4.3 (a) Vector A and the resultant vector after
multiplying A by a positive number 2.
(b) Vector A and resultant vectors after
multiplying it by a negative number –1
and –1.5.
(c) (d)
Fig. 4.4 (a) Vectors A and B. (b) Vectors A and B
added graphically. (c) Vectors B and A
added graphically. (d) Illustrating the
associative law of vector addition.
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PHYSICS 68
arranged head to tail, this graphical method is
called the headtotail method. The two vectors
and their resultant form three sides of a triangle,
so this method is also known as triangle method
of vector addition. If we find the resultant of
B + A as in Fig. 4.4(c), the same vector R is
obtained. Thus, vector addition is commutative:
A + B = B + A (4.1)
The addition of vectors also obeys the associative
law as illustrated in Fig. 4.4(d). The result of
adding vectors A and B first and then adding
vector C is the same as the result of adding B
and C first and then adding vector A :
(A + B) + C = A + (B + C) (4.2)
What is the result of adding two equal and
opposite vectors ? Consider two vectors A and
–A shown in Fig. 4.3(b). Their sum is A + (–A).
Since the magnitudes of the two vectors are the
same, but the directions are opposite, the
resultant vector has zero magnitude and is
represented by 0 called a null vector or a zero
vector :
A – A = 0 0= 0 (4.3)
Since the magnitude of a null vector is zero, its
direction cannot be specified.
The null vector also results when we multiply
a vector A by the number zero. The main
properties of 0 are :
A + 0 = A
? 0 = 0
0 A = 0 (4.4)
Fig. 4.5 (a) Two vectors A and B, – B is also shown. (b) Subtracting vector B from vector A – the result is R
2
. For
comparison, addition of vectors A and B, i.e. R
1
is also shown.
What is the physical meaning of a zero vector?
Consider the position and displacement vectors
in a plane as shown in Fig. 4.1(a). Now suppose
that an object which is at P at time t, moves to
P' and then comes back to P. Then, what is its
displacement? Since the initial and final
positions coincide, the displacement is a “null
vector”.
Subtraction of vectors can be defined in terms
of addition of vectors. We define the difference
of two vectors A and B as the sum of two vectors
A and –B :
A – B = A + (–B) (4.5)
It is shown in Fig 4.5. The vector –B is added to
vector A to get R
2
= (A – B). The vector R
1
= A + B
is also shown in the same figure for comparison.
We can also use the parallelogram method to
find the sum of two vectors. Suppose we have
two vectors A and B. To add these vectors, we
bring their tails to a common origin O as
shown in Fig. 4.6(a). Then we draw a line from
the head of A parallel to B and another line from
the head of B parallel to A to complete a
parallelogram OQSP. Now we join the point of
the intersection of these two lines to the origin
O. The resultant vector R is directed from the
common origin O along the diagonal (OS) of the
parallelogram [ Fig. 4.6(b)]. In Fig.4.6(c), the
triangle law is used to obtain the resultant of A
and B and we see that the two methods yield the
same result. Thus, the two methods are
equivalent.
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MOTION IN A PLANE 69
?
Example 4.1 Rain is falling vertically with
a speed of 35 m s
–1
. Winds starts blowing
after sometime with a speed of 12 m s
–1
in
east to west direction. In which direction
should a boy waiting at a bus stop hold
his umbrella ?
Fig. 4.7
Answer The velocity of the rain and the wind
are represented by the vectors v
r
and v
w
in Fig.
4.7 and are in the direction specified by the
problem. Using the rule of vector addition, we
see that the resultant of v
r
and v
w
is R as shown
in the figure. The magnitude of R is
R vv
r
2
w
2
= += + =

35 12 m s 37 m s
22 1 1
The direction ? that R makes with the vertical
is given by
12
tan 0.343
35
w
r
v
v
? == =
Or,
() ? = =° tan

.
1
0 343 19
Therefore, the boy should hold his umbrella
in the vertical plane at an angle of about 19
o
with the vertical towards the east.??????? ?
Fig. 4.6 (a) Two vectors A and B with their tails brought to a common origin. (b) The sum A + B obtained using
the parallelogram method. (c) The parallelogram method of vector addition is equivalent to the triangle
method.
4.5 RESOLUTION OF VECTORS
Let a and b be any two nonzero vectors in a
plane with different directions and let A be
another vector in the same plane(Fig. 4.8). A can
be expressed as a sum of two vectors — one
obtained by multiplying a by a real number and
the other obtained by multiplying b by another
real number. To see this, let O and P be the tail
and head of the vector A. Then, through O, draw
a straight line parallel to a, and through P, a
straight line parallel to b. Let them intersect at
Q. Then, we have
A = OP = OQ + QP (4.6)
But since OQ is parallel to a, and QP is parallel
to b, we can write :
OQ = ? a, and QP = µ b (4.7)
where ? and µ are real numbers.
Therefore, A = ? a + µ b (4.8)
Fig. 4.8 (a) Two noncolinear vectors a and b.
(b) Resolving a vector A in terms of vectors
a and b.
We say that A has been resolved into two
component vectors ? a and µ b along a and b
respectively. Using this method one can resolve
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