class 11 science

1. MOTION IN A
PLANE
CLASS XI

2. INTRODUCTION
The concepts of position, displacement, velocity and
acceleration that are needed to describe the motion of an
object along a straight line.
The directional aspect of these quantities can be taken care
of by + and – signs, as in one dimension only two directions
are possible.
In order to describe motion of an object in two dimensions
(a plane) or three dimensions (space), we need to use vectors.

3. SCALARS AND VECTORS

4. POSITION AND VECTOR
DISPLACEMENT
Position Vector: Position vector of an object at time t is the
position of the object relative to the origin. It is represented by
a straight line between the origin and the position at time t.
Position vector is used to specify the position of a certain
body.
Knowing the position of a body is vital when it comes to
describe the motion of that body.

5. The position vector of an object is
measured from the origin, in
general. Suppose an object is
placed in the space as shown:
So if an object is at a certain point P (say) at a certain
time, its position vector is given as described above.

6. Displacement Vector:
It is the vector which tells how much and in which direction
an object has changed its position in a given time interval.
B
A
O
Y
X
Consider that object is moving in
XY plane.
Object is at point A at any instant t
and at point B at any later instant
t’.
Displacement vector 𝐴𝐵 is the
displacement vector of the object in
time 𝑡 𝑡𝑜 𝑡′
Displacement
vector

7. Now, the displacement vector of the object from time interval 0
to t will be:
The displacement of the particle would be the vector line AB,
headed in the direction A to B.
The direction of displacement vector is always headed from
initial point to the final point.

8. VECTORS AND THEIR
NOTATIONS

9. Representation of Two-Dimensional Vectors
Two-dimensional vectors can be represented in three ways
as follows:
Geometric Representation of Vectors:
In this vector representation, we use an
arrow.
The length of the arrow denotes the
magnitude of the vector and the
direction of the vector is indicated by the
arrowhead.
The vector shown in the diagram can be written as 𝑂𝑄 and
its magnitude is written as |OQ|.

10. EQUAL VECTORS
Two vectors are said to be equal if they have the same
magnitude and same direction.
𝐴
𝐵
NEGATIVE OF A VECTOR
Two vectors are said to be negative vector having the same
magnitude but having opposite direction
𝐴
𝐵 = −𝐴

11. MODULIUS OF A VECTOR
Modulus of a vector means the length or the magnitude of
that vector. It is a scalar quantity.
𝒎𝒐𝒅𝒖𝒍𝒖𝒔 𝒐𝒇 𝒗𝒆𝒄𝒕𝒐𝒓 𝑨 = 𝑨 = 𝑨
UNIT VECTOR
A unit vector is a vector of unit vector drawn in the
direction of a given vector.
𝐴 =
𝐴
𝐴
=
𝐴
𝐴
Magnitude of a unit vector is a unity.
It gives direction of a vector.
It has no units and dimensions

12. FIXED VECTOR
The vector whose initial position is fixed is called fixed
vector or a localized vector
FREE VECTOR
The vector whose initial position is NOT fixed is called
FREE vector or a non-localized vector.
For example: the velocity vector of a particle moving along
a straight line is a free vector.

13. COLINEAR VECTOR
The vector which either act along the same line or along
parallel lines are called collinear vector.
Two vector having same direction (𝜽 = 𝟎°) are called like
or parallel vectors.
Two vector having opposite direction (𝜽 = 𝟏𝟖𝟎°) are called
unlike or antiparallel vectors.
𝐴 𝐵
𝐴
𝐵
𝐴
𝐵
𝐴
𝐵
(Like vectors)
(Unlike vectors)

14. COPLANAR VECTOR
The vector which act in the same plane are called coplanar
vector.
CO-INITIAL VECTOR
The vector which have the same initial point are called co-
initial vectors.

15. CO-TERMINUS VECTOR
The vector which have the common terminal point are called
co-terminus vector.

16. ZERO VECTOR
A zero vector or null vector is a vector that has a zero
magnitude and no direction.
Properties of Zero Vector (Null Vector)
When a zero vector is added to a non-zero vector, the
resultant vector is equal to the given non-zero vector.
𝑨 + 𝟎 = 𝑨
When a real number is multiplied by a zero vector, we get
zero vector.
𝝀𝟎 = 𝟎

17. When a vector is multiplied by zero, we get zero vector.
𝟎𝑨 = 𝟎
If 𝝀 and 𝝁 are two different non zero real number, then
relation
𝝀𝑨 = 𝝁𝑩
Can hold only if both 𝑨 and 𝑩 are zero vector.

18. Multiplication of Vectors with real
numbers
Multiplying a 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 :
𝝀𝑨 = 𝝀 𝑨 if 𝝀 > 𝑨
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.

19. Multiplying a 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

21. Addition and composition of Vectors
The resultant of two or more vectors is that single which
produces the same effect as the individual vectors together
would produce.
The process of adding two or more vectors is called
composition of vectors.
Vectors can be added geometrically.
Three laws of vector addition are
1. Triangle law of vector addition.
2. Parallelogram law of vector addition.
3. Polygon law of vector addition

22. 1. Triangle law of vector addition.
This law is used to add two vectors when the first vector's head is
joined to the tail of the second vector and then joining the tail of
the first vector to the head of the second vector to form a triangle,
and hence obtain the resultant sum vector.
It is also called the head-to-tail method for the addition of
vectors.
Two vectors P and Q as given and we
need to find their sum, then we can move
the vector Q in a way without changing
its magnitude and direction such its tail
is joined to the head of the vector P.
The sum of the vectors written as 𝑅 = 𝑃 + 𝑄

23. 2. Parallelogram law of vector addition.
The parallelogram law of vector addition is used to add two
vectors when the vectors that are to be added form the two
adjacent sides of a parallelogram by joining the tails of the two
vectors.
Then, the sum of the two vectors is given by the diagonal of the
parallelogram.
Two vectors 𝑃 and 𝑄 has been arranged in
such a way that they have common point
𝑂 and draw 𝑂𝐴 and 𝑂𝐵.
Complete parallelogram 𝑶𝑨𝑪𝑩.
Draw a diagonal 𝑂𝐶 which is resultant
vector.
𝑹 = 𝑷 + 𝑸

24. Vector addition is commutative
Two vectors 𝑃 𝑎𝑛𝑑 𝑄 be represented in
magnitude and direction by two sides
𝑂𝐴 𝑎𝑛𝑑 𝐴𝐵.
Complete parallelogram 𝑂𝐴𝐵𝐶 , such that
𝑂𝐴 = 𝐶𝐵 = 𝑃 𝑎𝑛𝑑 𝐴𝐵 = 𝑂𝐶 = 𝑄 then join OB.
In Δ𝑂𝐴𝐵 𝑂𝐴 + 𝐴𝐵 = 𝑂𝐵
(By triangle law of vector addition)
𝑷 + 𝑸 = 𝑹 … … . (𝟏)

25. In Δ𝑂𝐶𝐵 𝑂𝐶 + 𝐶𝐵 = 𝑂𝐵
(By triangle law of vector addition)
𝑸 + 𝑷 = 𝑹 … … . (𝟐)
From equation (1) and (2)
𝑷 + 𝑸 = 𝑸 + 𝑷

26. Vector addition is associative
The vectors 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 represented by 𝑃𝑄, 𝑄𝑅 𝑎𝑛𝑑 𝑅𝑆.
Now
𝒂 + 𝒃 = 𝑷𝑸 + 𝑸𝑹 = 𝑷𝑹
Also 𝒃 + 𝒄 = 𝑸𝑹 + 𝑹𝑺 = 𝑸𝑺
Hence 𝒂 + 𝒃 + 𝒄 = 𝑷𝑹 + 𝑹𝑺 = 𝑷𝑺
And 𝒂 + 𝒃 + 𝒄 = 𝑷𝑸 + 𝑸𝑺 = 𝑷𝑺
Therefore
𝒂 + 𝒃 + 𝒄 = 𝒂 + 𝒃 + 𝒄

27. Subtraction of vector
It is 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
The vector –B is added to vector A to get
𝑹𝟐 = (𝑨 − 𝑩)

28. RESOLUTION OF A VECTOR IN A
PLANE
It is the process of splitting a vector into two or more
vectors in such a way that their combined effect is same as
that of given vector. The vectors into which the given vector
is splitted are called components of vector.
A vector can be expressed in terms of
other vectors in the same plane.
If there are 3 vectors A, a and b, then A
can be expressed as sum of a and b
after multiplying them with some real
numbers

29. A can be resolved into two component vectors λa and μb.
Hence,
A = λa + μb.
Here λ and μ are real numbers.
This is called resolution of 𝑣𝑒𝑐𝑡𝑜𝑟 𝐴 in the direction of
𝑣𝑒𝑐𝑡𝑜𝑟 𝑎 and 𝑣𝑒𝑐𝑡𝑜𝑟 𝑏

30. RESOLUTION OF A VECTOR IN A
PLANE-RECTANGULAR
COMPONENTS
Consider the following vector r; the
vector r can be resolved into horizontal
and vertical components, these two
components add up to give us the
resultant vector i.e. vector r.
The horizontal component and the vertical component, the
horizontal component lies on the x-axis whereas the
vertical component lies on the y-axis,

31. The horizontal component will resemble
the shadow of the vector r falling on the
x-axis if the light were shining from
above.
Similarly, the vertical component will
resemble the shadow of vector r falling on
the y-axis if the light were shining from
the side.
𝐴𝑥
𝐴𝑦
𝐴
𝜃

32. RECTANGULAR COMPONENT OF
A VECTOR

35. ANALYSTICAL METHOD OF
VECTOR ADDITION
Magnitude of resultant 𝑹:
Draw SN perpendicular to OP produced.
∠𝑺𝑷𝑵 = ∠𝑸𝑶𝑷 = 𝜽, 𝑶𝑷 = 𝑨, 𝑷𝑺 = 𝑶𝑸 = 𝑩, 𝑶𝑺 = 𝑹
From right angled triangle Δ𝑆𝑁𝑃
𝑺𝑵
𝑷𝑺
= 𝐬𝐢𝐧 𝜽 𝒐𝒓 𝑺𝑵 = 𝑷𝑺 𝐬𝐢𝐧 𝜽 = 𝑩 𝐬𝐢𝐧 𝜽
𝑷𝑵
𝑷𝑺
= 𝐜𝐨𝐬 𝜽 𝒐𝒓 𝑷𝑵 = 𝑷𝑺 𝐜𝐨𝐬 𝜽 = 𝑩 𝐜𝐨𝐬 𝜽

36. Using Pythagoras theorem in right angled triangle ONS
𝑶𝑺𝟐
= 𝑶𝑵𝟐
+ 𝑺𝑵𝟐
= (𝑶𝑷 + 𝑷𝑵)𝟐
+𝑺𝑵𝟐
Or
𝑹𝟐 = (𝑨 + 𝑩 𝐜𝐨𝐬 𝜽)𝟐+(𝑩 𝐬𝐢𝐧 𝜽)𝟐
= 𝑨𝟐 + 𝑩𝟐 𝒄𝒐𝒔𝟐𝜽 + 𝒔𝒊𝒏𝟐𝜽 + 𝟐 𝑨𝑩 𝐜𝐨𝐬 𝜽
= 𝑨𝟐 + 𝑩𝟐 + 𝟐 𝑨𝑩 𝒄𝒐𝒔 𝜽
𝑹 = 𝑨𝟐 + 𝑩𝟐 + 𝟐 𝑨𝑩 𝒄𝒐𝒔 𝜽
Direction of resultant 𝑹:
𝑅 makes an angle 𝛽 with direction 𝐴.
From right angled triangle ONS
𝐭𝐚𝐧 𝜷 =
𝑺𝑵
𝑶𝑵
=
𝑺𝑵
𝑶𝑷 + 𝑷𝑵
𝒐𝒓 𝐭𝐚𝐧 𝜷 =
𝑩 𝐬𝐢𝐧 𝜽
𝑨 + 𝑩 𝐜𝐨𝐬 𝜽

37. SCALAR AND VECTOR
PRODUCTS OF VACTOR
SCALAR OR DOT PRODUCT OF TWO VECTORS
The scalar product of two vectors is equal to the
product of their magnitudes and the cosine of the
smaller angle between them. It is denoted by .
(dot).

38. To obtain scalar product,
From the figure

39. Properties of Scalar Product
(1) Commutative Law (2) Distributive law

41. VECTOR OR DOT PRODUCT OF TWO VECTORS
The vector product of two vectors is equal to the product of their
magnitudes and the sine of the smaller angle between them. It is
denoted by (cross).

42. Properties of vector Product

44. MOTION IN A PLANE
Position vector
The position vector r of a particle P located in
a plane with reference to the origin of an x-y
reference frame.
𝒓 = 𝒙𝒊 + 𝒚𝒋
where x and y are components of 𝒓 along x-,
and y- axes

45. (Displacement ∆r and
average velocity 𝒗of a
particle.)
A particle moves along the curve shown by
the thick line and is at P at time t and P′ at
time t′ .Then, the displacement is :
∆𝒓 = 𝒓′ − 𝒓
directed from P to P′.
In a component form can be written as
∆𝒓 = 𝒙′𝒊 + 𝒚𝒋 − 𝒙𝒊 + 𝒚𝒋 = 𝒊∆𝒙 + 𝒋∆𝒚
Displacement vector:

46. Velocity
The average velocity (v) of an object is the ratio of the
displacement and the corresponding time interval :
𝑣 =
∆𝑟
∆𝑡
=
∆𝒙𝒊 + ∆𝒚𝒋
∆𝑡
= 𝒊
∆𝒙
∆𝒕
+ 𝒋
∆𝒚
∆𝒕
𝒗 = 𝒗𝒙𝒊 + 𝒗𝒚𝒋
The direction of the average velocity is the same as that of ∆r
The velocity (instantaneous velocity) is given by the limiting
value of the average velocity as the time interval approaches
zero :
𝒗 = 𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒓
∆𝒕
=
𝒅𝒓
𝒅𝒕

47. The meaning of the limiting process can be easily understood
with the help of Fig
In these figures, the thick line represents the path of an
object,
Object position P at time t. P1 , P2 and P3 represent the
positions of the object after times ∆t1 ,∆t2 , and ∆t3 .
∆r1 , ∆r2 , and ∆r3 are the displacements of the object in times
∆t1 , ∆t2 , and ∆t3

48. 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
𝒗 =
𝒅𝒓
𝒅𝒕
∴ 𝒗 = 𝐥𝐢𝐦
∆𝒕→𝟎
𝒊
∆𝒙
∆𝒕
+ 𝒋
∆𝒚
∆𝒕
∴ 𝒗= 𝒊𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒙
∆𝒕
+ 𝒋 𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒚
∆𝒕
∴ 𝒗= 𝒊
𝒅𝒙
∆𝒕
+ 𝒋
𝒅𝒚
∆𝒕
= 𝒗𝒙 𝒊 + 𝒗𝒚 𝒋
𝒘𝒉𝒆𝒓 𝒗𝒙 =
𝒅𝒙
∆𝒕
, 𝒗𝒚 =
𝒅𝒚
∆𝒕

49. If the expressions for the coordinates x and y are known as
functions of time, we can use these equations to 𝒗𝒙 find 𝒗𝒚 and
The magnitude of 𝒗 is then
𝒗 = 𝒗𝟐
𝒙 + 𝒗𝟐
𝒚
and the direction of v is given by the angle θ
𝐭𝐚𝐧 𝜽 =
𝒗𝒙
𝒗𝒚
, 𝜽 = 𝒕𝒂𝒏−𝟏 𝒗𝒙
𝒗𝒚

50. 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
∆𝒕
=
∆(𝒗𝒙𝒊 + 𝒗𝒚𝒋)
∆𝒕
=
∆𝒗𝒙
∆𝒕
𝒊 +
∆𝒗𝒚
∆𝒕
𝒋
∴ 𝒂 = 𝒂𝒙𝒊 + 𝒂𝒚𝒋
The acceleration (instantaneous acceleration) is the limiting
value of the average acceleration as the time interval
approaches zero :

51. ∴ 𝒂 = 𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒗
∆𝒕
∴ ∆𝒗 = 𝒗𝒙 𝒊 + 𝒗𝒚 𝒋
∴ 𝒂= 𝒊𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒗𝒙
∆𝒕
+ 𝒋 𝐥𝐢𝐦
∆𝒕→𝟎
∆𝒚
∆𝒕
∴ 𝒂 = 𝒂𝒙 𝒊 + 𝒂𝒚 𝒋
𝒘𝒉𝒆𝒓𝐞 𝐚𝐱 =
𝐝𝐯𝐱
𝐝𝐭
, 𝐚𝐲 =
𝐝𝐯𝐲
𝐝𝐭

52. Graphically limiting process has been shown in figure.
𝑃 represents the position of the object at time t and
𝑃1 , 𝑃2 , 𝑃3 positions after time ∆𝑡 , ∆𝑡 2 , ∆𝑡 3 , respectively (∆t
1 > ∆t 2 >∆t 3 ).
The velocity vectors at points 𝑃, 𝑃1 , 𝑃2 , 𝑃3 are also shown in
Figs

53. In each case of ∆𝑡, ∆𝑣 is obtained using the triangle law of
vector addition.
Direction of average acceleration is the same as that of ∆𝑣.
∆t decreases, the direction of ∆v changes and consequently,
the direction of the acceleration changes.

54. Finally, in the limit ∆𝑡 → 0 the average acceleration becomes
the instantaneous acceleration and has the direction as
shown.

55. 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.
Let
velocity of the object be 𝑣0 at time 𝑡 = 0.
𝑣 at time 𝑡.
Then as per definition
𝒂 =
𝒗 − 𝒗𝟎
𝒕 − 𝟎
=
𝒗 − 𝒗𝟎
𝒕
∴ 𝒗 = 𝒗𝟎 + 𝒂𝒕

56. In terms of components :
𝒗𝒙 = 𝒗𝟎𝒙 + 𝒂𝒙𝒕
𝒗𝒚 = 𝒗𝟎𝒚 + 𝒂𝒚𝒕
Changes in position vector 𝒓 with time.
𝐿𝑒𝑡 𝑎𝑡
𝑡 = 0, position vector 𝑟0, velocity 𝑣0
𝑡 = 𝑡, position vector 𝑟, velocity 𝑣
𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 =
𝒗𝟎 + 𝒗
𝟐
The displacement is the average velocity multiplied by the
time interval :
𝒓 − 𝒓𝟎 =
𝒗𝟎 + 𝒗
𝟐
𝒕

57. ∴ 𝒓 − 𝒓𝟎 =
(𝒗𝟎 + 𝒂𝒕) + 𝒗𝟎
𝟐
𝒕
∴ 𝒓 − 𝒓𝟎 =
𝟐𝒗𝟎 + 𝒂𝒕
𝟐
𝒕
∴ 𝒓 − 𝒓𝟎 = 𝒗𝟎𝒕 +
𝒂𝒕𝟐
𝟐
∴ 𝒓 = 𝒓𝟎 + 𝒗𝟎𝒕 +
𝒂𝒕𝟐
𝟐
Above equation in can be written as
𝒙 = 𝒙𝟎 + 𝒗𝟎𝒙𝒕 +
𝟏
𝟐
𝒂𝒙𝒕𝟐
, 𝐲 = 𝒚𝟎 + 𝒗𝟎𝒚𝒕 +
𝟏
𝟐
𝒂𝒚 𝒕𝟐
Motion in a plane (two-dimensions) can be treated as two separate
simultaneous one-dimensional motions with constant acceleration along
two perpendicular directions.

58. PROJECTILE MOTION
When an object is in flight after being projected or thrown then
that object is called projectile and this motion is called
projectile motion.
Example: motion of cricket ball, baseball(Resistance of air is
neglected)
It is a result of two separate simultaneously occurring
components of motion.
1.Horizontal component without any acceleration.(no force in
this direction)
2.Vertical component with constant acceleration due to force of
gravity.

59. Projectile is projected with velocity
𝑣0 and makes an angle 𝜃0 with X-
axis.
Acceleration acting on the projectile is
due to gravity which is directed
vertically downward
…………….(1)
For initial velocity 𝑣0
…………….(2)
…………….(3)

60. For the initial position
Using equation (2) and (3)
…………….(4)
…………….(5)
The component of velocity at any time t can be obtained from
…………….(7)
…………….(6)

61. Equation of trajectory of a
projectile
From equation (4)
…………….(8)
𝒙 = 𝒗𝟎𝒄𝒐𝒔𝜽𝟎 𝒕
𝒕 =
𝒙
𝒗𝟎𝒄𝒐𝒔𝜽𝟎
Put value of above equation in (5)

62. In equation (8),quantities θ0,g and v0 are all constants and equation (8)
can be compared with the equation
𝒚 = 𝒂𝒙 − 𝒃𝒙𝟐
where 𝑎 and 𝑏 are constants
This equation 𝒚 = 𝒂𝒙 − 𝒃𝒙𝟐 is the equation of the parabola.
From this we conclude that path of the projectile is a parabola as shown
in figure

63. Time taken to achieve maximum height
Let maximum height be 𝐻, maximum time is 𝒕𝒎
When projectile attains maximum height, the y component of its
velocity(𝑣𝑦) becomes zero.
So equation (7) 𝒗𝒚 = 𝒗𝟎𝒔𝒊𝒏𝜽𝟎 − 𝒈𝒕 becomes
…………….(9)

64. Maximum Height
substitute
Into equation
𝒚 = (𝒗𝟎𝒔𝒊𝒏𝜽𝟎𝒕)𝒕 −
𝟏
𝟐
𝒈𝒕𝟐

65. Time of flight
Substituting 𝒚 = 𝟎 and 𝒕 = 𝒕𝒇 in equation
𝒚 = (𝒗𝟎𝒔𝒊𝒏𝜽𝟎𝒕)𝒕 −
𝟏
𝟐
𝒈𝒕𝟐

66. Range of projectile
Substitute 𝑥 = 𝑅 and 𝒕 = 𝒕𝒇 in equation (4)

67. UNIFORM CIRCULAR MOTION
When an object moves in a circular path at a constant
speed then motion of the object is called uniform
circular motion.
If the speed of motion is constant for a particle moving
in a circular motion still the particles accelerates
because of constantly changing direction of the velocity.
Here in circular motion ,we use angular velocity in
place of velocity we used while studying linear motion

68. (A) Centripetal Acceleration
When a particle moves along a
circular path it must have components
of acceleration perpendicular to the
path when its speed is constant. Such
type of circular motion is called
Uniform circular motion .
Let us consider the particle moves in
a circular path with constant speed.

69. This figure represent a particle
that is moving in circular path of
radius R and have its centre at O.
Here vectors v1 and v2 represents
the velocities at points P and Q
respectively.
Figure given below shows the
vector change in velocity that
is Δv as the particle moves
from P to Q in time Δt.

70. In circular motion, an acceleration acts on the body,
whose direction is always towards the centre of the path.
This acceleration is called centripetal acceleration.

71. Angular Displacement:
Angular displacement is the angle subtended by the
position vector at the centre of the circular path.
Angular displacement (Δθ) = (ΔS/r)
where Δs is the linear displacement and r is the radius.
Its unit is radian.

72. Angular Velocity:
The time rate of change of angular displacement (Δθ) is
called angular velocity.
Angular velocity (ω) = (Δθ/Δt)
Angular velocity is a vector quantity and its unit is
rad/s.
Relation between linear velocity (v) and angular velocity
(ω) is given by
v = rω

73. Angular Acceleration:
The time rate of change of angular velocity (dω) is called
angular acceleration.
Its unit is 𝑟𝑎𝑑
𝑠2 and dimensional formula is [𝑇−2
].
Relation between linear acceleration (a) and angular acceleration
(α).
a = rα
where, r = radius