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 daily-life 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 proper
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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PHYSICS66
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 λ
gives a vector whose magnitude is changed by
the factor λ but the direction is the same as that
of A :
λ A = λ A if λ > 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
−λ gives another vector whose direction is
opposite to the direction of A and whose
magnitude is λ 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 λ by which a vector A is multiplied
could be a scalar having its own physical
dimension. Then, the dimension of λ A is the
product of the dimensions of λ 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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PHYSICS68
arranged head to tail, this graphical method is
called the head-to-tail 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
λ 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
t
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 v v
r
2
w
2
= + = + =
− −
35 12 m s 37 m s
2 2 1 1
The direction
θ
that R makes with the vertical
is given by
12
tan 0.343
35
w
r
v
v
θ
= = =
Or,
( )
θ
= = °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. t
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 non-zero 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 =
λ
a, and QP = µ b (4.7)
where
λ
and µ are real numbers.
Therefore, A =
λ
a + µ b (4.8)
Fig. 4.8 (a) Two non-colinear 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
λ
a and
µ
b along a and b
respectively. Using this method one can resolve
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PHYSICS70
Fig. 4.9 (a) Unit vectors
i
,
j
and
k
lie along the x-, y-, and z-axes. (b) A vector A is resolved into its
components A
x
and A
y
along x-, and y- axes. (c) A
1
and A
2
expressed in terms of
i
and
j
.
a given vector into two component vectors along
a set of two vectors – all the three lie in the same
plane. It is convenient to resolve a general vector
along the axes of a rectangular coordinate
system using vectors of unit magnitude. These
are called unit vectors that we discuss now. A
unit vector is a vector of unit magnitude and
points in a particular direction. It has no
dimension and unit. It is used to specify a
direction only. Unit vectors along the x-, y- and
z-axes of a rectangular coordinate system are
denoted by
i
,
j
and
ˆ
k
, respectively, as shown
in Fig. 4.9(a).
Since these are unit vectors, we have
ˆ
i
=
ˆ
j
=
ˆ
k
=1 (4.9)
These unit vectors are perpendicular to each
other. In this text, they are printed in bold face
with a cap (^) to distinguish them from other
vectors. Since we are dealing with motion in two
dimensions in this chapter, we require use of
only two unit vectors. If we multiply a unit vector,
say
ˆ
n
by a scalar, the result is a vector
λλ
λλ
λ
= λ
ˆ
n
. In general, a vector A can be written as
A = |A|
ˆ
n
(4.10)
where
ˆ
n
is a unit vector along A.
We can now resolve a vector A in terms
of component vectors that lie along unit vectors
i
ˆ
and
j
. Consider a vector A that lies in x-y
plane as shown in Fig. 4.9(b). We draw lines from
the head of A perpendicular to the coordinate
axes as in Fig. 4.9(b), and get vectors A
1
and A
2
such that A
1
+ A
2
= A. Since A
1
is parallel to
i
and A
2
is parallel to
j
, we have :
A
1
= A
x
i
, A
2
= A
y
j
(4.11)
where A
x
and A
y
are real numbers.
Thus, A = A
x
i
+ A
y
j
(4.12)
This is represented in Fig. 4.9(c). The quantities
A
x
and A
y
are called x-, and y- components of the
vector A. Note that A
x
is itself not a vector, but
A
x
i
is a vector, and so is A
y
j
. Using simple
trigonometry, we can express A
x
and A
y
in terms
of the magnitude of A and the angle
θ
it makes
with the x-axis :
A
x
= A cos
θ
A
y
= A sin
θ
(4.13)
As is clear from Eq. (4.13), a component of a
vector can be positive, negative or zero
depending on the value of
θ
.
Now, we have two ways to specify a vector A
in a plane. It can be specified by :
(i) its magnitude A and the direction θ it makes
with the x-axis; or
(ii) its components A
x
and A
y
If A and θ are given, A
x
and A
y
can be obtained
using Eq. (4.13). If A
x
and A
y
are given, A and
θ
can be obtained as follows :
A A A A
x
2
y
2 2 2 2 2
+ = +cos sin
θ θ
= A
2
Or,
A A A
x
2
y
2
= +
(4.14)
And
tan , tanθ θ= =
−
A
A
A
A
y
x
y
x
1
(4.15)
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MOTION IN A PLANE 71
t
B i j= +B B
x y
Let R be their sum. We have
R = A + B
(
)
(
)
= + + +A A B B
x y x y
i j i j
(4.19a)
Since vectors obey the commutative and
associative laws, we can arrange and regroup
the vectors in Eq. (4.19a) as convenient to us :
(
)
(
)
R i j= + + +A B A B
x x y y
(4.19b)
Since
R i j= +R R
x y
(4.20)
we have,
x x x y y y
R A B , R A B
= + = +
(4.21)
Thus, each component of the resultant
vector R is the sum of the corresponding
components of A and B.
In three dimensions, we have
A i j k= + +A A A
x y z
B i j k= + +B B B
x y z
R A B i j k= + = + +R R R
x y z
with
R A B
x x x
=
+
R A B
y y y
=
+
R A B
z z z
=
+
(4.22)
This method can be extended to addition and
subtraction of any number of vectors. For
example, if vectors a, b and c are given as
a i j k= + +a a a
x y z
b i j k= + +b b b
x y z
c i j k= + +c c c
x y z
(4.23a)
then, a vector T = a + b – c has components :
T
a
b
c
x
x
x
x
=
+
−
T a b c
y y y y
=
+
−
(4.23b)
T a b c
z z z z
=
+
−
.
Example 4.2 Find the magnitude and
direction of the resultant of two vectors A
and B in terms of their magnitudes and
angle
θ
between them.
Fig. 4.9 (d) A vector A resolved into components along
x-, y-, and z-axes
* Note that angles
α
,
β
, and
γ
are angles in space. They are between pairs of lines, which are not coplanar.
So far we have considered a vector lying in
an x-y plane. The same procedure can be used
to resolve a general vector A into three
components along x-, y-, and z-axes in three
dimensions. If
α
,
β
, and
γ
are the angles*
between A and the x-, y-, and z-axes, respectively
[Fig. 4.9(d)], we have
(d)
x y z
A A cos , A A cos , A A cos
α β γ
= = =
(4.16a)
In general, we have
ˆ ˆ ˆ
x y z
A A A= + +
A i j k
(4.16b)
The magnitude of vector A is
2 2
2
x y z
A A A A
= + +
(4.16c)
A position vector r can be expressed as
r i j k= + +x y z
(4.17)
where x, y, and z are the components of r along
x-, y-, z-axes, respectively.
4.6 VECTOR ADDITION – ANALYTICAL
METHOD
Although the graphical method of adding vectors
helps us in visualising the vectors and the
resultant vector, it is sometimes tedious and has
limited accuracy. It is much easier to add vectors
by combining their respective components.
Consider two vectors A and B in x-y plane with
components A
x
, A
y
and B
x
, B
y
:
A i j= +A A
x y
(4.18)
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PHYSICS72
t
Fig. 4.10
Answer Let OP and OQ represent the two vectors
A and B making an angle
θ
(Fig. 4.10). Then,
using the parallelogram method of vector
addition, OS represents the resultant vector R :
R = A + B
SN is normal to OP and PM is normal to OS.
From the geometry of the figure,
OS
2
= ON
2
+ SN
2
but ON = OP + PN = A + B cos
θ
SN = B sin
θ
OS
2
= (A + B cos
θ
)
2
+ (B sin
θ
)
2
or, R
2
= A
2
+ B
2
+ 2AB cos
θ
R A B 2AB
2 2
= + + cos
θ
(4.24a)
In ∆ OSN, SN = OS sin
α
= R sin
α
, and
in ∆ PSN, SN = PS sin
θ
= B sin
θ
Therefore, R sin
α
= B sin
θ
or,
R B
sin sin
θ α
=
(4.24b)
Similarly,
PM = A sin
α
= B sin
β
or,
A B
sin sin
β α
=
(4.24c)
Combining Eqs. (4.24b) and (4.24c), we get
R A
sin sin sin
θ β α
= =
B
(4.24d)
Using Eq. (4.24d), we get:
sin sin
α θ
=
B
R
(4.24e)
where R is given by Eq. (4.24a).
or,
sin
tan
cos
SN B
OP PN A B
θ
α
θ
= =
+ +
(4.24f)
Equation (4.24a) gives the magnitude of the
resultant and Eqs. (4.24e) and (4.24f) its direction.
Equation (4.24a) is known as the Law of cosines
and Eq. (4.24d) as the Law of sines. t
Example 4.3 A motorboat is racing
towards north at 25 km/h and the water
current in that region is 10 km/h in the
direction of 60° east of south. Find the
resultant velocity of the boat.
Answer The vector v
b
representing the velocity
of the motorboat and the vector v
c
representing
the water current are shown in Fig. 4.11 in
directions specified by the problem. Using the
parallelogram method of addition, the resultant
R is obtained in the direction shown in the
figure.
Fig. 4.11
We can obtain the magnitude of R using the Law
of cosine :
R v v v v=
b
2
c
2
b c
2 cos120+ +
o
= 25 10 2 25 10 -1/2 22 km/h
2 2
+ + × ×
( )
≅
To obtain the direction, we apply the Law of sines
R v
c
sin sin
θ φ
=
or,
sin
φ θ
=
v
R
c
sin
=
10 sin120
21.8
10 3
2
21.8
0.397
×
=
×
≅
φ
≅ 23.4
t
4.7 MOTION IN A PLANE
In this section we shall see how to describe
motion in two dimensions using vectors.
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MOTION IN A PLANE 73
4.7.1 Position Vector and Displacement
The position vector r of a particle P located in a
plane with reference to the origin of an x-y
reference frame (Fig. 4.12) is given by
r i j= +x y
where x and y are components of r along x-, and
y- axes or simply they are the coordinates of
the object.
(a)
(b)
Fig. 4.12 (a) Position vector r. (b) Displacement ∆r and
average velocity v of a particle.
Suppose a particle moves along the curve shown
by the thick line and is at P at time t and P′ at
time t′ [Fig. 4.12(b)]. Then, the displacement is :
∆r = r′ – r (4.25)
and is directed from P to P′ .
We can write Eq. (4.25) in a component form:
∆r
(
)
(
)
= + − +x' y' x
i j i jy
= +
i j∆ ∆x y
where ∆x = x ′ – x, ∆y = y′ – y (4.26)
Velocity
The average velocity
(
)
v
of an object is the ratio
of the displacement and the corresponding time
interval :
v
r i j
i j= =
+
= +
∆
∆
∆ ∆
∆
∆
∆
∆
∆
t
x y
t
x
t
y
t
(4.27)
Or,
ˆ
x y
v v= +
v i j
Since
v
r
=
∆
∆
t
, the direction of the average velocity
is the same as that of ∆r (Fig. 4.12). The velocity
(instantaneous velocity) is given by the limiting
value of the average velocity as the time interval
approaches zero :
v
r r
= =
→
lim
t t
t∆
∆
∆
0
d
d
(4.28)
The meaning of the limiting process can be easily
understood with the help of Fig 4.13(a) to (d). In
these figures, the thick line represents the path
of an object, which is at P at time t. P
1
, P
2
and
P
3
represent the positions of the object after
times ∆t
1
,∆t
2
, and ∆t
3
. ∆r
1
, ∆r
2
, and ∆r
3
are the
displacements of the object in times ∆t
1
, ∆t
2
, and
Fig. 4.13 As the time interval ∆t approaches zero, the average velocity approaches the velocity v. The direction
of
v
is parallel to the line tangent to the path.
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PHYSICS74
∆t
3
, respectively. The direction of the average
velocity
v
is shown in figures (a), (b) and (c) for
three decreasing values of ∆t, i.e. ∆t
1
,∆t
2
, and ∆t
3
,
(∆t
1
> ∆t
2
> ∆t
3
). As ∆t
→→
→→
→ 0, ∆r
→→
→→
→ 0
and is along the tangent to the path [Fig. 4.13(d)].
Therefore, the direction of velocity at any point
on the path of an object is tangential to the
path at that point and is in the direction of
motion.
We can express v in a component form :
v
r
=
d
d
t
= +
→
lim
x
t
y
tt∆
∆
∆
∆
∆0
i j
(4.29)
= +
→
→
i jlim
x
t
lim
y
t
t t
∆
∆
∆
∆
∆
∆
0 0
Or,
v i j i j= + = +
d
d
d
d
x
t
y
t
v v
x y
.
where
v
x
t
v
y
t
x y
= =
d
d
d
d
,
(4.30a)
So, if the expressions for the coordinates x and
y are known as functions of time, we can use
these equations to find v
x
and v
y
.
The magnitude of v is then
v v v
x
2
y
2
= +
(4.30b)
and the direction of v is given by the angle
θ
:
tan tan
1
θ θ
= =
−
v
v
v
v
y
x
y
x
,
(4.30c)
v
x
, v
y
and angle
θ
are shown in Fig. 4.14 for a
velocity vector v at point p.
Acceleration
The average acceleration a of an object for a
time interval ∆t moving in x-y plane is the change
in velocity divided by the time interval :
(
)
a
v
i j
i j= =
+
= +
∆
∆
∆
∆
∆
∆
∆
∆
t
v v
t
v
t
v
t
x y
x
y
(4.31a)
Or,
a i j= +a a
x y
. (4.31b)
The acceleration (instantaneous acceleration)
is the limiting value of the average acceleration
as the time interval approaches zero :
a
v
=
→
lim
t
t
∆
∆
∆
0
(4.32a)
Since
∆ ∆ ∆v = +v v ,
x y
i j
we have
a i j= +
→
→
lim
v
t
lim
v
t
t
x
t
y
∆
∆
∆
∆
∆
∆
0 0
Or,
a i j= +a a
x y
(4.32b)
where,
a
v
t
, a
v
t
x
x
y
y
= =
d
d
d
d
(4.32c)*
As in the case of velocity, we can understand
graphically the limiting process used in defining
acceleration on a graph showing the path of the
object’s motion. This is shown in Figs. 4.15(a) to
(d). P represents the position of the object at
time t and P
1
, P
2
, P
3
positions after time ∆t
1
, ∆t
2
,
∆t
3
, respectively (∆t
1
> ∆t
2
>∆t
3
). The velocity vectors
at points P, P
1
, P
2
, P
3
are also shown in Figs. 4.15
(a), (b) and (c). In each case of ∆t, ∆v is obtained
using the triangle law of vector addition.
By
definition, the direction of average acceleration
is the same as that of ∆v. We see that as ∆t
decreases, the direction of ∆v changes and
consequently, the direction of the acceleration
changes. Finally, in the limit ∆t g0 [Fig. 4.15(d)],
the average acceleration becomes the
instantaneous acceleration and has the direction
as shown.
Fig. 4.14 The components v
x
and v
y
of velocity v and
the angle
θ
it makes with x-axis. Note that
v
x
= v cos
θ
, v
y
= v sin
θ
.
* In terms of x and y, a
x
and a
y
can be expressed as
2020-21
MOTION IN A PLANE 75
t
x (m)
Note that in one dimension, the velocity and
the acceleration of an object are always along
the same straight line (either in the same
direction or in the opposite direction).
However, for motion in two or three
dimensions, velocity and acceleration vectors
may have any angle between 0° and 180°
between them.
Example 4.4 The position of a particle is
given by
r i j k = + +3.0t
ˆ
.
ˆ
.
ˆ
2 0 5 0
2
t
where t is in seconds and the coefficients
have the proper units for r to be in metres.
(a) Find v(t) and a(t) of the particle. (b) Find
the magnitude and direction of v(t) at
t = 1.0 s.
Answer
( )
(
)
v
r
i j kt
t
t
t t
2
= = + +
d
d
d
d
3.0 2.0 5.0
= +3.0 .0
i j4 t
( )
a
v
jt
t
=
d
d
= +4.0
a = 4.0 m s
–2
along y- direction
At t = 1.0 s,
ˆ ˆ
3.0 4.0
v = i + j
It’s magnitude is
2 2
1
-
= 3 4 5.0 m s
v + =
and direction is
-1 1
4
= tan tan 53
3
y
x
v
v
θ
−
°
= ≅
with x-axis.
t
4.8 MOTION IN A PLANE WITH CONSTANT
ACCELERATION
Suppose that an object is moving in x-y plane
and its acceleration a is constant. Over an
interval of time, the average acceleration will
equal this constant value. Now, let the velocity
of the object be v
0
at time t = 0 and v at time t.
Then, by definition
a
v
v
v
v
0 0
=
−
−
=
−
t
t
0
Or,
v
v
a
0
=
+
t
(4.33a)
In terms of components :
v
v
a
t
x ox x
=
+
v
v
a
t
y oy y
=
+
(4.33b)
Let us now find how the position r changes with
time. We follow the method used in the one-
dimensional case. Let r
o
and r be the position
vectors of the particle at time 0 and t and let the
velocities at these instants be v
o
and v. Then,
over this time interval t, the average velocity is
(v
o
+ v)/2. The displacement is the average
velocity multiplied by the time interval :
r r
v v
v a v
0
0
0 0
− =
+
=
+
( )
+
2 2
t
t
t
Fig. 4.15 The average acceleration for three time intervals (a) ∆t
1
, (b) ∆t
2
, and (c) ∆t
3
, (∆t
1
> ∆t
2
> ∆t
3
). (d) In the
limit ∆t g0, the average acceleration becomes the acceleration.
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PHYSICS76
t
t
2
1
2
t t
= +
0
v a
Or,
r r v a
0 0
= + +t t
1
2
2
(4.34a)
It can be easily verified that the derivative of
Eq. (4.34a), i.e.
d
d
r
t
gives Eq.(4.33a) and it also
satisfies the condition that at t=0, r = r
o
.
Equation (4.34a) can be written in component
form as
x x v t a t
ox x
= + +
0
2
1
2
2
1
2
0 oy y
y y v t a t
= + +
(4.34b)
One immediate interpretation of Eq.(4.34b) is that
the motions in x- and y-directions can be treated
independently of each other. That is, motion in
a plane (two-dimensions) can be treated as two
separate simultaneous one-dimensional
motions with constant acceleration along two
perpendicular directions. This is an important
result and is useful in analysing motion of objects
in two dimensions. A similar result holds for three
dimensions. The choice of perpendicular
directions is convenient in many physical
situations, as we shall see in section 4.10 for
projectile motion.
Example 4.5 A particle starts from origin
at t = 0 with a velocity 5.0 î m/s and moves
in x-y plane under action of a force which
produces a constant acceleration of
(3.0i
$$
$$
$
+2.0j
$$
$$
$
) m/s
2
. (a) What is the
y
-coordinate of the particle at the instant
its x-coordinate is 84 m ? (b) What is the
speed of the particle at this time ?
Answer From Eq. (4.34a) for r
0
= 0, the position
of the particle is given by
(
)
2
1
2
t t t
= +
0
r v a
(
)
(
)
2
ˆ ˆ ˆ
5.0 1/2 3.0 2.0
t t
= + +i i j
(
)
2 2
ˆ ˆ
5.0 1.5 1.0
t t t
= + +
i j
Therefore,
(
)
2
5.0 1.5
x t t t
= +
(
)
2
1.0
y t t
= +
Given x (t) = 84 m, t = ?
5.0 t + 1.5 t
2
= 84
⇒⇒
⇒⇒
⇒ t = 6 s
At t = 6 s, y = 1.0 (6)
2
= 36.0 m
Now, the velocity
( )
d
ˆ ˆ
5.0 3.0 2.0
d
t t
t
= = + +
r
v i j
At t = 6 s,
v i j= +23.
0 12.0
speed
2 2 1
23 12 26 m s
−
= = + ≅v
. t
4.9 RELATIVE VELOCITY IN TWO
DIMENSIONS
The concept of relative velocity, introduced in
section 3.7 for motion along a straight line, can
be easily extended to include motion in a plane
or in three dimensions. Suppose that two objects
A and B are moving with velocities v
A
and v
B
(each with respect to some common frame of
reference, say ground.). Then, velocity of object
A relative to that of B is :
v
AB
= v
A
– v
B
(4.35a)
and similarly, the velocity of object B relative to
that of A is :
v
BA
= v
B
–
v
A
Therefore, v
AB
= – v
BA
(4.35b)
and,
v v
AB BA
=
(4.35c)
Example 4.6 Rain is falling vertically with
a speed of 35 m s
–1
. A woman rides a bicycle
with a speed of 12 m s
–1
in east to west
direction. What is the direction in which
she should hold her umbrella ?
Answer In Fig. 4.16 v
r
represents the velocity
of rain and v
b
, the velocity of the bicycle, the
woman is riding. Both these velocities are with
respect to the ground. Since the woman is riding
a bicycle, the velocity of rain as experienced by
Fig. 4.16
her is the velocity of rain relative to the velocity
of the bicycle she is riding. That is v
rb
= v
r
–
v
b
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MOTION IN A PLANE 77
This relative velocity vector as shown in
Fig. 4.16 makes an angle
θ
with the vertical. It is
given by
tan
12
35
0.343
θ
= = =
v
v
b
r
Or,
θ
≅
19
Therefore, the woman should hold her
umbrella at an angle of about 19° with the
vertical towards the west.
Note carefully the difference between this
Example and the Example 4.1. In Example 4.1,
the boy experiences the resultant (vector
sum) of two velocities while in this example,
the woman experiences the velocity of rain
relative to the bicycle (the vector difference
of the two velocities). t
4.10 PROJECTILE MOTION
As an application of the ideas developed in the
previous sections, we consider the motion of a
projectile. An object that is in flight after being
thrown or projected is called a projectile. Such
a projectile might be a football, a cricket ball, a
baseball or any other object. The motion of a
projectile may be thought of as the result of two
separate, simultaneously occurring components
of motions. One component is along a horizontal
direction without any acceleration and the other
along the vertical direction with constant
acceleration due to the force of gravity. It was
Galileo who first stated this independency of the
horizontal and the vertical components of
projectile motion in his Dialogue on the great
world systems (1632).
In our discussion, we shall assume that the
air resistance has negligible effect on the motion
of the projectile. Suppose that the projectile is
launched with velocity v
o
that makes an angle
θ
o
with the x-axis as shown in Fig. 4.17.
After the object has been projected, the
acceleration acting on it is that due to gravity
which is directed vertically downward:
a j= −g
Or, a
x
= 0, a
y
= – g (4.36)
The components of initial velocity v
o
are :
v
ox
= v
o
cos
θ
o
v
oy
= v
o
sin
θ
o
(4.37)
If we take the initial position to be the origin of
the reference frame as shown in Fig. 4.17, we
have :
x
o
= 0, y
o
= 0
Then, Eq.(4.34b) becomes :
x = v
ox
t = (v
o
cos
θ
o
) t
and y = (v
o
sin
θ
o
) t – ( ½ )g t
2
(4.38)
The components of velocity at time t can be
obtained using Eq.(4.33b) :
v
x
= v
ox
= v
o
cos
θ
o
v
y
= v
o
sin
θ
o
– g t (4.39)
Equation (4.38) gives the x-, and y-coordinates
of the position of a projectile at time t in terms of
two parameters — initial speed v
o
and projection
angle
θ
o
. Notice that the choice of mutually
perpendicular x-, and y-directions for the
analysis of the projectile motion has resulted in
a simplification. One of the components of
velocity, i.e. x-component remains constant
throughout the motion and only the
y- component changes, like an object in free fall
in vertical direction. This is shown graphically
at few instants in Fig. 4.18. Note that at the point
of maximum height, v
y
= 0 and therefore,
θ
= =
−
tan 0
1
v
v
y
x
Equation of path of a projectile
What is the shape of the path followed by the
projectile? This can be seen by eliminating the
time between the expressions for x and y as
given in Eq. (4.38). We obtain:
Fig 4.17 Motion of an object projected with velocity
v
o
at angle
θ
0
.
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PHYSICS78
t
t
( )
( )
2
o
2
o o
tan
2 cos
g
y x x
v
θ
θ
= −
(4.40)
Now, since g,
θ
o
and v
o
are constants, Eq. (4.40)
is of the form y = a x + b x
2
, in which a and b are
constants. This is the equation of a parabola,
i.e. the path of the projectile is a parabola
(Fig. 4.18).
Fig. 4.18 The path of a projectile is a parabola.
Time of maximum height
How much time does the projectile take to reach
the maximum height ? Let this time be denoted
by t
m
. Since at this point, v
y
= 0, we have from
Eq. (4.39):
v
y
= v
o
sin
θ
o
– g t
m
= 0
Or, t
m
= v
o
sin
θ
o
/g (4.41a)
The total time T
f
during which the projectile is
in flight can be obtained by putting y = 0 in
Eq. (4.38). We get :
T
f
= 2 (v
o
sin
θ
o
)/g (4.41b)
T
f
is known as the time of flight of the projectile.
We note that T
f
= 2 t
m
, which is expected
because of the symmetry of the parabolic path.
Maximum height of a projectile
The maximum height h
m
reached by the
projectile can be calculated by substituting
t = t
m
in Eq. (4.38) :
( )
y h v
v
g
g v
g
m 0
0 0
= =
−
sin
sin
2
sin
0
0 0
2
θ
θ θ
Or,
(
)
h
v
m
0
=
sin
0
θ
2
2g
(4.42)
Horizontal range of a projectile
The horizontal distance travelled by a projectile
from its initial position (x = y = 0) to the position
where it passes y = 0 during its fall is called the
horizontal range, R. It is the distance travelled
during the time of flight T
f
. Therefore, the range
R is
R = (v
o
cos
θ
o
) (T
f
)
=(v
o
cos
θ
o
) (2 v
o
sin
θ
o
)/g
Or,
R
v
g
0
2
=
sin 2
0
θ
(4.43a)
Equation (4.43a) shows that for a given
projection velocity v
o
, R is maximum when sin
2
θ
0
is maximum, i.e., when
θ
0
= 45
0
.
The maximum horizontal range is, therefore,
R
v
g
m
0
2
=
(4.43b)
Example 4.7 Galileo, in his book Two new
sciences, stated that “for elevations which
exceed or fall short of 45° by equal
amounts, the ranges are equal”. Prove this
statement.
Answer For a projectile launched with velocity
v
o
at an angle
θ
o
, the range is given by
0
sin2
2
0
v
R
g
θ
=
Now, for angles, (45° +
α
) and ( 45°
–
α
), 2
θ
o
is
(90° + 2
α
) and ( 90° – 2
α
) , respectively. The
values of sin (90° + 2
α
) and sin (90° – 2
α
) are
the same, equal to that of cos 2
α
. Therefore,
ranges are equal for elevations which exceed or
fall short of 45° by equal amounts
α
. t
Example 4.8 A hiker stands on the edge
of a cliff 490 m above the ground and
throws a stone horizontally with an initial
speed of 15 m s
-1
. Neglecting air resistance,
find the time taken by the stone to reach
the ground, and the speed with which it
hits the ground. (Take g = 9.8 m s
-2
).
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MOTION IN A PLANE 79
t
Answer We choose the origin of the x-,and y-
axis at the edge of the cliff and t = 0 s at the
instant the stone is thrown. Choose the positive
direction of x-axis to be along the initial velocity
and the positive direction of y-axis to be the
vertically upward direction. The x-, and y-
components of the motion can be treated
independently. The equations of motion are :
x (t) = x
o
+ v
ox
t
y (t) = y
o
+ v
oy
t +(1/2) a
y
t
2
Here, x
o
= y
o
= 0, v
oy
= 0, a
y
=
–g = –9.8 m s
-2
,
v
ox
= 15 m s
-1
.
The stone hits the ground when y(t) = – 490 m.
– 490 m = –(1/2)(9.8) t
2
.
This gives t =10 s.
The velocity components are v
x
= v
ox
and
v
y
= v
oy
– g t
so that when the stone hits the ground :
v
ox
= 15 m s
–1
v
oy
= 0 – 9.8 × 10 = – 98 m s
–1
Therefore, the speed of the stone is
2 2
15 98 99 m s
2 2 1
x y
v v
−
+ = + =
t
Example 4.9 A cricket ball is thrown at a
speed of 28 m s
–1
in a direction 30° above
the horizontal. Calculate (a) the maximum
height, (b) the time taken by the ball to
return to the same level, and (c) the
distance from the thrower to the point
where the ball returns to the same level.
Answer (a) The maximum height is given by
(
)
( )
(
)
2
2
o
sin
28 sin 30
m
2 2 9.8
0
m
v
h
g
θ
°
= =
=
×
×
=
14
14
2
9.8
10.0 m
(b) The time taken to return to the same level is
T
f
= (2 v
o
sin
θ
o
)/g = (2× 28 × sin 30°
)/9.8
= 28/9.8 s = 2.9 s
(c) The distance from the thrower to the point
where the ball returns to the same level is
R
(
)
2
o o
sin2
28 28 sin 60
69 m
9.8
o
v
g
θ
× ×
= = =
t
4.11 UNIFORM CIRCULAR MOTION
When an object follows a circular path at a
constant speed, the motion of the object is called
uniform circular motion. The word “uniform”
refers to the speed, which is uniform (constant)
throughout the motion. Suppose an object is
moving with uniform speed v in a circle of radius
R as shown in Fig. 4.19. Since the velocity of the
object is changing continuously in direction, the
object undergoes acceleration. Let us find the
magnitude and the direction of this acceleration.
Neglecting air resistance - what does
the assumption really mean?
While treating the topic of projectile motion,
we have stated that we assume that the
air resistance has no effect on the motion
of the projectile. You must understand what
the statement really means. Friction, force
due to viscosity, air resistance are all
dissipative forces. In the presence of any of
such forces opposing motion, any object will
lose some part of its initial energy and
consequently, momentum too. Thus, a
projectile that traverses a parabolic path
would certainly show deviation from its
idealised trajectory in the presence of air
resistance. It will not hit the ground with
the same speed with which it was projected
from it. In the absence of air resistance, the
x-component of the velocity remains
constant and it is only the y-component that
undergoes a continuous change. However,
in the presence of air resistance, both of
these would get affected. That would mean
that the range would be less than the one
given by Eq. (4.43). Maximum height
attained would also be less than that
predicted by Eq. (4.42). Can you then,
anticipate the change in the time of flight?
In order to avoid air resistance, we will
have to perform the experiment in vacuum
or under low pressure, which is not easy.
When we use a phrase like ‘neglect air
resistance’, we imply that the change in
parameters such as range, height etc. is
much smaller than their values without air
resistance. The calculation without air
resistance is much simpler than that with
air resistance.
2020-21
PHYSICS80
Let r and r′ be the position vectors and v and
v′ the velocities of the object when it is at point P
and P′ as shown in Fig. 4.19(a). By definition,
velocity at a point is along the tangent at that
point in the direction of motion. The velocity
vectors v and v
′ are as shown in Fig. 4.19(a1).
∆v is obtained in Fig. 4.19 (a2) using the triangle
law of vector addition. Since the path is circular,
v is perpendicular to r
and so is v′ to r′.
Therefore, ∆v is perpendicular to ∆r. Since
average acceleration is along ∆v
a
v
=
∆
∆t
, the
average acceleration
a
is perpendicular to ∆r. If
we place ∆v on the line that bisects the angle
between r and r′, we see that it is directed towards
the centre of the circle. Figure 4.19(b) shows the
same quantities for smaller time interval. ∆v and
hence
a
is again directed towards the centre.
In Fig. 4.19(c), ∆t
0 and the average
acceleration becomes the instantaneous
acceleration. It is directed towards the centre*.
Thus, we find that the acceleration of an object
in uniform circular motion is always directed
towards the centre of the circle. Let us now find
the magnitude of the acceleration.
The magnitude of a is, by definition, given by
a
v
=
→∆
∆
∆
t 0
t
Let the angle between position vectors r and
r′ be ∆
θ
. Since the velocity vectors v and v′ are
always perpendicular to the position vectors, the
angle between them is also ∆
θ
. Therefore, the
triangle CPP′ formed by the position vectors and
the triangle GHI formed by the velocity vectors
v, v′ and ∆v are similar (Fig. 4.19a). Therefore,
the ratio of the base-length to side-length for
one of the triangles is equal to that of the other
triangle. That is :
∆
∆
v
r
v
R
=
Or,
∆
∆
v
r
= v
R
Therefore,
a
v r r
=
→
=
→
=
→∆
∆
∆
∆
∆
∆
∆
∆
∆
t 0 0
R R
0
t
t
v
t
v
t
t
If ∆t is small, ∆θ will also be small and then arc
PP
′
can be approximately taken to be|∆r|:
∆ ∆r ≅ v t
∆
∆
r
t
v≅
Or,
∆
∆
∆
t 0
t
v
→
=
r
Therefore, the centripetal acceleration a
c
is :
Fig. 4.19 Velocity and acceleration of an object in uniform circular motion. The time interval ∆t decreases from
(a) to (c) where it is zero. The acceleration is directed, at each point of the path, towards the centre of
the circle.
* In the limit ∆t0, ∆r becomes perpendicular to r. In this limit ∆v
→
0 and is consequently also perpendicular
to V. Therefore, the acceleration is directed towards the centre, at each point of the circular path.
lim
limlim lim
lim
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MOTION IN A PLANE 81
t
a
c
=
v
R
v = v
2
/R (4.44)
Thus, the acceleration of an object moving with
speed v in a circle of radius R has a magnitude
v
2
/R and is always directed towards the centre.
This is why this acceleration is called centripetal
acceleration (a term proposed by Newton). A
thorough analysis of centripetal acceleration was
first published in 1673 by the Dutch scientist
Christiaan Huygens (1629-1695) but it was
probably known to Newton also some years earlier.
“Centripetal” comes from a Greek term which means
‘centre-seeking’. Since v and R are constant, the
magnitude of the centripetal acceleration is also
constant. However, the direction changes —
pointing always towards the centre. Therefore, a
centripetal acceleration is not a constant vector.
We have another way of describing the
velocity and the acceleration of an object in
uniform circular motion. As the object moves
from P to P′ in time ∆t (= t′ – t), the line CP
(Fig. 4.19) turns through an angle ∆
θ
as shown
in the figure. ∆
θ
is called angular distance. We
define the angular speed
ω
(Greek letter omega)
as the time rate of change of angular
displacement :
ω
θ
=
∆
∆
t
(4.45)
Now, if the distance travelled by the object
during the time ∆t is ∆s, i.e. PP
′
is ∆s, then :
v
s
t
=
∆
∆
but ∆s = R ∆
θ
. Therefore :
v R
t
= =
∆
∆
θ
ω
R
v = R
ω
(4.46)
We can express centripetal acceleration a
c
in
terms of angular speed :
a
v
R
R
R
R
c
= = =
2
2
2
ω
ω
2
a R
c
=
ω
2
(4.47)
The time taken by an object to make one revolution
is known as its time period T and the number of
revolution made in one second is called its
frequency ν (=1/T). However, during this time the
distance moved by the object is s = 2πR.
Therefore, v = 2πR/T =2πRν (4.48)
In terms of frequency ν, we have
ω
= 2πν
v = 2πRν
a
c
= 4π
2
ν
2
R (4.49)
Example 4.10 An insect trapped in a
circular groove of radius 12 cm moves along
the groove steadily and completes 7
revolutions in 100 s. (a) What is the
angular speed, and the linear speed of the
motion? (b) Is the acceleration vector a
constant vector ? What is its magnitude ?
Answer This is an example of uniform circular
motion. Here R = 12 cm. The angular speed ω is
given by
ω
= 2π/T = 2π × 7/100 = 0.44 rad/s
The linear speed v is :
v =ω R = 0.44 s
-1
× 12 cm = 5.3 cm s
-1
The direction of velocity v is along the tangent
to the circle at every point. The acceleration is
directed towards the centre of the circle. Since
this direction changes continuously,
acceleration here is not a constant vector.
However, the magnitude of acceleration is
constant:
a =
ω
2
R = (0.44 s
–1
)
2
(12 cm)
= 2.3 cm s
-2
t
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PHYSICS82
SUMMARY
1. Scalar quantities are quantities with magnitudes only. Examples are distance, speed,
mass and temperature.
2. Vector quantities are quantities with magnitude and direction both. Examples are
displacement, velocity and acceleration. They obey special rules of vector algebra.
3. A vector A multiplied by a real number λ is also a vector, whose magnitude is λ times
the magnitude of the vector A and whose direction is the same or opposite depending
upon whether λ is positive or negative.
4. Two vectors A and B may be added graphically using head-to-tail method or parallelogram
method.
5. Vector addition is commutative :
A + B = B + A
It also obeys the associative law :
(A + B) + C = A + (B + C)
6. A null or zero vector is a vector with zero magnitude. Since the magnitude is zero, we
don’t have to specify its direction. It has the properties :
A + 0 = A
λ0 = 0
0 A = 0
7. The subtraction of vector B from A is defined as the sum of A and –B :
A – B = A+ (–B)
8. A vector A can be resolved into component along two given vectors a and b lying in the
same plane :
A =
λ
a +
µ
b
where λ and µ are real numbers.
9. A unit vector associated with a vector A has magnitude 1 and is along the vector A:
ˆ
n
A
A
=
The unit vectors
i, j, k
are vectors of unit magnitude and point in the direction of
the x-, y-, and z-axes, respectively in a right-handed coordinate system.
10. A vector A can be expressed as
A i + j= A A
x y
where A
x
, A
y
are its components along x-, and y -axes. If vector A makes an angle
θ
with the x-axis, then A
x
= A cos
θ
, A
y
=A sin
θ
and
2 2
, tan = .
y
x y
x
A
A A A
A
θ
= = +A
11. Vectors can be conveniently added using analytical method. If sum of two vectors A
and B, that lie in x-y plane, is R, then :
R i j= +R R
x y
, where, R
x
= A
x
+ B
x
, and R
y
= A
y
+ B
y
12. The position vector of an object in x-y plane is given by
r = i jx y
+
and the
displacement from position r to position r’ is given by
∆r = r′− r
=
′
− +
′
−( ) ( ) x x y y
i j
=
∆ + ∆x y
i j
13. If an object undergoes a displacement ∆r in time ∆t, its average velocity is given by
v =
∆
∆
r
t
. The velocity of an object at time t is the limiting value of the average velocity
2020-21
MOTION IN A PLANE 83
as ∆t tends to zero :
v =
∆
∆
∆
=
t → 0
r r
t t
d
d
. It can be written in unit vector notation as :
v i j k= + +v v v
x y z
where
t
z
v
t
y
v
t
x
v
zyx
d
d
=,
d
d
= ,
d
d
=
When position of an object is plotted on a coordinate system, v is always tangent to
the curve representing the path of the object.
14. If the velocity of an object changes from v to v′in time ∆t, then its average acceleration
is given by:
a
v v' v
=
−
=
∆
∆
∆
t
t
The acceleration a at any time t is the limiting value of
a
as ∆t
0 :
a
v v
=
→
=
∆
∆
∆
t
t t
0
d
d
In component form, we have :
a i j k= + +a a a
x y z
where,
a
dv
dt
, a
dv
dt
, a
dv
dt
x
x
y
y
z
z
= = =
15. If an object is moving in a plane with constant acceleration
2 2
=
x y
a a a
= +a
and
its position vector at time t = 0 is r
o
, then at any other time t, it will be at a point given
by:
2
1
2
t t
= + +
o o
r r v a
and its velocity is given by :
v = v
o
+ a t
where v
o
is the velocity at time t = 0
In component form :
2
1
2
o ox x
x x v t a t
= + +
2
1
2
o oy y
y y v t a t
= + +
v v a t
x
ox
x
= +
v v a t
y oy y
= +
Motion in a plane can be treated as superposition of two separate simultaneous one-
dimensional motions along two perpendicular directions
16. An object that is in flight after being projected is called a projectile. If an object is
projected with initial velocity v
o
making an angle
θ
o
with x-axis and if we assume its
initial position to coincide with the origin of the coordinate system, then the position
and velocity of the projectile at time t are given by :
x = (v
o
cos
θ
o
) t
y = (v
o
sin
θ
o
) t − (1/2) g t
2
v
x
= v
ox
= v
o
cos
θ
o
v
y
= v
o
sin
θ
o
− g t
The path of a projectile is parabolic and is given by :
( )
( )
2
0
2
tan
cos
o o
gx
y x –
2 v
θ
θ
=
The maximum height that a projectile attains is :
lim
lim
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PHYSICS84
h
v
2g
m
o o
=
( )
sin
2
q
The time taken to reach this height is :
g
v
t
oo
m
θ
sin
=
The horizontal distance travelled by a projectile from its initial position to the position
it passes y = 0 during its fall is called the range, R of the projectile. It is :
2
sin 2
o
o
v
R
g
θ
=
17. When an object follows a circular path at constant speed, the motion of the object is
called uniform circular motion. The magnitude of its acceleration is a
c
= v
2
/R. The
direction of a
c
is always towards the centre of the circle.
The angular speed
ω
, is the rate of change of angular distance. It is related to velocity
v by v =
ω
R. The acceleration is a
c
=
ω
2
R.
If T is the time period of revolution of the object in circular motion and ν is its
frequency, we have
ω
= 2π
ν, v = 2πνR, a
c
= 4π
2
ν
2
R
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MOTION IN A PLANE 85
POINTS TO PONDER
1. The path length traversed by an object between two points is, in general, not the same as
the magnitude of displacement. The displacement depends only on the end points; the
path length (as the name implies) depends on the actual path. The two quantities are
equal only if the object does not change its direction during the course of motion. In all
other cases, the path length is greater than the magnitude of displacement.
2. In view of point 1 above, the average speed of an object is greater than or equal to the
magnitude of the average velocity over a given time interval. The two are equal only if the
path length is equal to the magnitude of displacement.
3. The vector equations (4.33a) and (4.34a) do not involve any choice of axes. Of course,
you can always resolve them along any two independent axes.
4. The kinematic equations for uniform acceleration do not apply to the case of uniform
circular motion since in this case the magnitude of acceleration is constant but its
direction is changing.
5. An object subjected to two velocities v
1
and v
2
has a resultant velocity v = v
1
+ v
2
. Take
care to distinguish it from velocity of object 1 relative to velocity of object 2 : v
12
= v
1
− v
2
.
Here v
1
and v
2
are velocities with reference to some common reference frame.
6. The resultant acceleration of an object in circular motion is towards the centre only if
the speed is constant.
7. The shape of the trajectory of the motion of an object is not determined by the acceleration
alone but also depends on the initial conditions of motion ( initial position and initial
velocity). For example, the trajectory of an object moving under the same acceleration
due to gravity can be a straight line or a parabola depending on the initial conditions.
EXERCISES
4.1 State, for each of the following physical quantities, if it is a scalar or a vector :
volume, mass, speed, acceleration, density, number of moles, velocity, angular
frequency, displacement, angular velocity.
4.2 Pick out the two scalar quantities in the following list :
force, angular momentum, work, current, linear momentum, electric field, average
velocity, magnetic moment, relative velocity.
4.3 Pick out the only vector quantity in the following list :
Temperature, pressure, impulse, time, power, total path length, energy, gravitational
potential, coefficient of friction, charge.
4.4 State with reasons, whether the following algebraic operations with scalar and vector
physical quantities are meaningful :
(a) adding any two scalars, (b) adding a scalar to a vector of the same dimensions ,
(c) multiplying any vector by any scalar, (d) multiplying any two scalars, (e) adding any
two vectors, (f) adding a component of a vector to the same vector.
4.5 Read each statement below carefully and state with reasons, if it is true or false :
(a) The magnitude of a vector is always a scalar, (b) each component of a vector is
always a scalar, (c) the total path length is always equal to the magnitude of the
displacement vector of a particle. (d) the average speed of a particle (defined as total
path length divided by the time taken to cover the path) is either greater or equal to
the magnitude of average velocity of the particle over the same interval of time, (e)
Three vectors not lying in a plane can never add up to give a null vector.
4.6 Establish the following vector inequalities geometrically or otherwise :
(a) |a+b|
<
|a| + |b|
(b) |a+b|
>
||a|
−−
−−
−|b||
2020-21
PHYSICS86
Fig. 4.20
(c) |a
−−
−−
−b|
<
|a| + |b|
(d) |a
−−
−−
−b|
>
||a|
−−
−−
− |b||
When does the equality sign above apply?
4.7 Given a + b + c + d = 0, which of the following statements
are correct :
(a) a, b, c, and d must each be a null vector,
(b) The magnitude of (a + c) equals the magnitude of
( b + d),
(c) The magnitude of a can never be greater than the
sum of the magnitudes of b, c, and d,
(d) b + c must lie in the plane of a and d if a and d are
not collinear, and in the line of a and d, if they are
collinear ?
4.8 Three girls skating on a circular ice ground of radius
200 m start from a point P on the edge of the ground
and reach a point Q diametrically opposite to P following
different paths as shown in Fig. 4.20. What is the
magnitude of the displacement vector for each ? For
which girl is this equal to the actual length of
path skate ?
4.9 A cyclist starts from the centre O of a circular park of radius 1 km, reaches the edge P
of the park, then cycles along the circumference, and returns to the centre along QO
as shown in Fig. 4.21. If the round trip takes 10 min, what is the (a) net displacement,
(b) average velocity, and (c) average speed of the cyclist ?
Fig. 4.21
4.10 On an open ground, a motorist follows a track that turns to his left by an angle of 60
0
after every 500 m. Starting from a given turn, specify the displacement of the motorist
at the third, sixth and eighth turn. Compare the magnitude of the displacement with
the total path length covered by the motorist in each case.
4.11 A passenger arriving in a new town wishes to go from the station to a hotel located
10 km away on a straight road from the station. A dishonest cabman takes him along
a circuitous path 23 km long and reaches the hotel in 28 min. What is (a) the average
speed of the taxi, (b) the magnitude of average velocity ? Are the two equal ?
4.12 Rain is falling vertically with a speed of 30 m s
-1
. A woman rides a bicycle with a speed
of 10 m s
-1
in the north to south direction. What is the direction in which she should
hold her umbrella ?
4.13 A man can swim with a speed of 4.0 km/h in still water. How long does he take to
cross a river 1.0 km wide if the river flows steadily at 3.0 km/h and he makes his
Q
2020-21
MOTION IN A PLANE 87
strokes normal to the river current? How far down the river does he go when he
reaches the other bank ?
4.14 In a harbour, wind is blowing at the speed of 72 km/h and the flag on the mast of a boat
anchored in the harbour flutters along the N-E direction. If the boat starts moving at a
speed of 51 km/h to the north, what is the direction of the flag on the mast of the boat?
4.15 The ceiling of a long hall is 25 m high. What is the maximum horizontal distance that
a ball thrown with a speed of 40 m s
-1
can go without hitting the ceiling of the hall ?
4.16 A cricketer can throw a ball to a maximum horizontal distance of 100 m. How much
high above the ground can the cricketer throw the same ball ?
4.17 A stone tied to the end of a string 80 cm long is whirled in a horizontal circle with a
constant speed. If the stone makes 14 revolutions in 25 s, what is the magnitude and
direction of acceleration of the stone ?
4.18 An aircraft executes a horizontal loop of radius 1.00 km with a steady speed of 900
km/h. Compare its centripetal acceleration with the acceleration due to gravity.
4.19 Read each statement below carefully and state, with reasons, if it is true or false :
(a) The net acceleration of a particle in circular motion is always along the radius of
the circle towards the centre
(b) The velocity vector of a particle at a point is always along the tangent to the path
of the particle at that point
(c) The acceleration vector of a particle in uniform circular motion averaged over one
cycle is a null vector
4.20 The position of a particle is given by
2
ˆ ˆ ˆ
3.0 2.0 4.0 m
t t= − +
r i j k
where t is in seconds and the coefficients have the proper units for r to be in metres.
(a) Find the v and a of the particle? (b) What is the magnitude and direction of
velocity of the particle at t = 2.0 s ?
4.21 A particle starts from the origin at t = 0 s with a velocity of 10.0
j
m/s and moves in
the x-y plane with a constant acceleration of
(
)
8.0 2.0
i j+
m s
-2
. (a) At what time is
the x- coordinate of the particle 16 m? What is the y-coordinate of the particle at
that time? (b) What is the speed of the particle at the time ?
4.22
i
and
j
are unit vectors along x- and y- axis respectively. What is the magnitude
and direction of the vectors
i j+
, and
i j−
? What are the components of a vector
A=
2
i j+ 3
along the directions of
i j+
and
i j−
? [You may use graphical method]
4.23 For any arbitrary motion in space, which of the following relations are true :
(a) v
average
= (1/2) (v (t
1
) + v (t
2
))
(b) v
average
= [r(t
2
) - r(t
1
) ] /(t
2
– t
1
)
(c) v (t) = v (0) + a t
(d) r (t) = r (0) + v (0) t + (1/2) a t
2
(e) a
average
=[ v (t
2
) - v (t
1
)] /( t
2
– t
1
)
(The ‘average’ stands for average of the quantity over the time interval t
1
to t
2
)
4.24 Read each statement below carefully and state, with reasons and examples, if it is
true or false :
A scalar quantity is one that
(a) is conserved in a process
(b) can never take negative values
(c) must be dimensionless
(d) does not vary from one point to another in space
(e) has the same value for observers with different orientations of axes.
4.25 An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at
a ground observation point by the aircraft positions 10.0 s apart is 30°, what is the
speed of the aircraft ?
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PHYSICS88
Additional Exercises
4.26 A vector has magnitude and direction. Does it have a location in space ? Can it vary
with time ? Will two equal vectors a and b at different locations in space necessarily
have identical physical effects ? Give examples in support of your answer.
4.27 A vector has both magnitude and direction. Does it mean that anything that has
magnitude and direction is necessarily a vector ? The rotation of a body can be specified
by the direction of the axis of rotation, and the angle of rotation about the axis. Does
that make any rotation a vector ?
4.28 Can you associate vectors with (a) the length of a wire bent into a loop, (b) a plane
area, (c) a sphere ? Explain.
4.29 A bullet fired at an angle of 30° with the horizontal hits the ground 3.0 km away. By
adjusting its angle of projection, can one hope to hit a target 5.0 km away ? Assume
the muzzle speed to be fixed, and neglect air resistance.
4.30 A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/h passes
directly overhead an anti-aircraft gun. At what angle from the vertical should the gun
be fired for the shell with muzzle speed 600 m s
-1
to hit the plane ? At what minimum
altitude should the pilot fly the plane to avoid being hit ? (Take g = 10 m s
-2
).
4.31 A cyclist is riding with a speed of 27 km/h. As he approaches a circular turn on the
road of radius 80 m, he applies brakes and reduces his speed at the constant rate of
0.50 m/s every second. What is the magnitude and direction of the net acceleration of
the cyclist on the circular turn ?
4.32 (a) Show that for a projectile the angle between the velocity and the x-axis as a function
of time is given by
( )
−
ox
0y
v
gtv
=
1-
tantθ
(b) Shows that the projection angle θ
0
for a projectile launched from the origin is
given by
0
R
4h
=
m
1-
tanθ
where the symbols have their usual meaning.
2020-21
