ch-3

1. UNIT – II- KINEMATICS
CHAPTER – 4
MOTION IN A PLANE

2. TOPIC TO BE DISCUSSED
• Introduction
• Scalar and vector quantities
• Vectors and their notations; equality of vectors,
multiplication of vectors by a real number; addition and
subtraction of vectors, relative velocity, Unit vector;
• Resolution of a vector in a plane-rectangular components,
• Scalar and Vector product of vectors.
• Motion in a plane, cases of uniform velocity and uniform
acceleration
• Projectile motion
• Uniform circular motion.

3. INTRODUCTION
• To describe motion of an object in two
dimensions (a plane) or three dimensions
(space), we need to use vectors.
• Then we 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 it in detail
the projectile motion.
• Circular motion which is a familiar class of
motion that has a special significance in
daily-life situations.

4. SCALARS AND VECTORS
QUANTITIES
• Scalar Quantities: The physical quantities which
are completely specified by their magnitude or size
alone are called scalar quantities.
Examples: Length, mass, density, speed, work, etc.
• Vector Quantities: The physical quantities which
are characterised by both magnitude and direction.
Examples: Velocity, displacement, acceleration,
force, momentum, torque etc.

5. CHARACTERISTICS OF
VECTORS
(iii)
(iv)
(i) It possess both magnitude and direction.
(ii) It does not obey the ordinary laws of
Algebra.
It changes either magnitude or direction
or both magnitude and direction.
The vectors are represented by bold
letters or letters having arrow over them.

6. POSITION AND DISPLACEMENT
VECTORS
(a) Position (OP or OP’) and displacement (PP’)vectors.
(b)Displacement vector PQ and different courses of
motion. The displacement vector is the same PQ for
different paths of journey, say PABCQ, PDQ, and PBEFQ.

7. EQUALITY OF VECTORS
• Two vectors A and B are said to be equal
if, they have the same magnitude and the
same direction.
(a) Two equal vectors A and B.
(b) Two vectors A′ and B′ are unequal though they are of the same
length.

8. 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 same as
that of A.
• Note:

9. ADDITION OF VECTORS —
GRAPHICAL METHOD
(a) Vectors A and B. (b) Vectors A and B added graphically.
(c) Vectors B and A added graphically.
In this procedure of vector addition, vectors are arranged
head to tail, this graphical method is called the head-to-tail
method.

10. PROPERTIES OF VECTOR
ADDITION
• vector addition obey Commutative law.
• vector addition also obeys Associative
law.

11. TYPES OF VECTOR ADDITION
• Triangle Law
If two sides of a triangle completely represent two
vectors both in magnitude and direction taken in same
order, then the third side taken in opposite order
represents the resultant of the two vectors both in
magnitude and direction.

12. TYPES OF VECTOR ADDITION
• Parallelogram Law
If two vectors are represented both in magnitude and
direction by the adjacent sides of a parallelogram drawn
from a point, then the resultant vector is represented
both in magnitude and direction by the diagonal of the
parallelogram passing through the same point.

13. TYPES OF VECTOR ADDITION
• Polygon Law
If a number of vectors are represented both in magnitude and
direction by the sides of a polygon taken in the same order, then the
resultant vector is represented both in magnitude and direction by
closing side of the polygon taken in the opposite order.

14. SUBTRACTION OF VECTORS
• The difference of two vectors A and B as the sum of
two vectors A and –B :
(a) Two vectors A and B, – B is also shown.
(b) Subtracting vector B from vector A – the result is R2. For comparison, addition of
vectors A and B, i.e. R1 is also shown.

15. ZERO OR NULL VECTOR
• If the magnitude of the two vectors are same, but
opposite in direction, then the resultant vector
has zero magnitude wthout any specific
directionand is called a null vector or a zero
vector.
• The main properties of 0 are :

16. TRIANGLE LAW OF VECTOR ADDITION
• If two sides of a triangle completely represent two
vectors both in magnitude and direction taken in
same order, then the third side taken in opposite
order represents the resultant of the two vectors
both in magnitude and direction.
In ∆ O C M,
O C 2 = O M2
C M2
O C 2
+
= (OA+AM)2 +
C M2
OC 2 = OA2 + 2 OA×
2 2
OC2 = OA2 + 2OA×AM +AC2
P
Q
R
θ
O A
c
M

17. AM = AC cos θ
OC2 = OA2 + 2OA × AM + AC2
OC2 = OA2 + 2OA × AC cos θ+ AC2
R2 = P2 + 2P × Q cos θ + Q2
R2 =P2 + 2 PQ cos θ + Q2
R=(P2 + 2PQ cos θ + Q2) 1 /2
P
R
Q
θ
M
In ∆CAM,
cos θ = AM/AC
O A
c

18. DIRECTION OF RESULTANT VECTOR
In triangle CAM;
sin θ = C M /AC
cos θ = AM / AC
In triangle OCM;
tan α= C M/ OM
= C M/ (OA+ AM)
= AC sin θ / (P+
AC c o s θ)
tan α = Q sin θ / (P+ Q
P
Q
R
θ
O A
c
M
α

19. PARALLELOGRAM LAW OF VECTOR
ADDITION
If two vectors are represented in
direction and magnitude by two
adjacent sides of parallelogram then
the resultant vector is given in
magnitude and direction by the
diagonal of the parallelogram
starting from the common point of
the adjacent sides.
R =A+ B

20. The diagram above shows two
vectors A and B with angle p between
them.
R is the resultant ofAand B
R =A+ B
This is the resultant in vector
R is the magnitude of vector R
SimilarlyAand B are the magnitudes
of vectors A and B
R = √(A2 + B2 +2ABCos p) or
[A2 + B2 +2ABCos p]1/2
To give the direction of R we find
the angle q that R makes with B
Tan q = (ASin p)/(B +ACos q)

21. UNIT VECTORS
• 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-andz-axes of
a rectangular coordinate system are
denoted by iˆ , jˆ and kˆ respectively
In general, a vector A can be written as
where n is a unit vector along A.

22. RESOLUTION OF A VECTOR IN A PLANE
(Rectangular Components of vectors)
• Consider a vector A that lies in x-y plane as shown in Fig.
• From fig, we have
• where Ax and Ay are real numbers. Thus
• The quantities Ax and Ay are called x-, and y-
components of the vector A.
•Using simple trigonometry, we can express Ax
and Ay in terms of the magnitude of A and the angle
θit makes with the x- axis :
•its magnitude A and the direction θ it makes with the x-axis;

23. RESOLUTION OF A VECTOR IN
3-DIMENSION
• If α
, β
, and γare the angles between A
and the x-, y-, and z-axes, respectively,
we have
• where x, y, and z are the components of r along x-, y-,
z-axes, respectively.

24. SCALAR OR DOT PRODUCT OF VECTORS
• Scalar product of two vectors is defined as the product of
the magnitude of two vectors with cosine of smaller
angle between them.

25. VECTOR OR CROSS PRODUCT OF VECTORS
Vector product of two vectors is defined as the product of the
magnitude of two vectors with sine of smaller angle between them
with unit vector perpendicular to those two vectors.
• θis angle between A and B taken in anti-clockwise direction.
• is unit vector in the direction perpendicular to the plane
containing A and B.
Note: In Chapter-7, vector product will be discussed in detail.

26. 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 is given
by
• where x and y are components of r along
x-, and y- axes or simply they are the
coordinates of the object.
•Suppose 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 :
and is directed from P to P
′.

27. VELOCITY
• The average velocity (v) of an object is the ratio of the
displacement and the corresponding time interval :
The velocity (instantaneous velocity) is
given by the limiting value of the
average velocity as the time interval
approaches zero :

28. As the time
interval Δt
approaches zero,
the average
velocity
approaches the
velocity v. The
direction
of v is parallel to
the line tangent
to the path.

29. ACCELERATION
• The average acceleration a of an object for a time interval Δ
t moving
in x-y plane is the change in velocity divided by the time interval :
• The acceleration (instantaneous acceleration) is the limiting
value of the average acceleration as the time interval approaches
zero :

30. The average acceleration for three time intervals (a)
Δt1, (b) Δt2, and (c) Δt3, (Δt1> Δt2> Δt3). (d) In the
limit Δt 0, the average acceleration becomes the
acceleration.

31. MOTION IN A PLANE WITH CONSTANT
ACCELERATION
Expression for velocity in a plane
• Suppose that an object is moving in x-y plane
and its acceleration a is constant. Now, let the
velocity of the object be v0 at time t = 0 and v at time
t. Then, by definition
• In terms of components :

32. Expression for Displacement in a plane
• Let ro and r be the position vectors of the particle at time 0
and t and let the velocities at these instants be vo and v.The
displacement is the average velocity multiplied by the time
interval :
• In component form as,
Note: Motion in a plane (two-dimensions) can be treated as two separate simultaneous
one-dimensional motions with constant acceleration along two perpendicular
directions.

33. RELATIVE VELOCITY IN TWO
DIMENSIONS
• Suppose that two objects A and B are moving with
velocities vA and vB (each with respect to some common
frame of reference, say ground.). Then, velocity of object
A relative to that of B is :
and similarly, the velocity of object B relative to that of A
is :

34. PROBLEM ON RELATIVE VELOCITY IN A
PLANE
Rain is falling vertically with a speed of 35 m/s. A woman rides
a bicycle with a speed of 12 m/s in east to west direction.
What is the direction in which she should hold her umbrella ?
• Since the woman is riding a bicycle, the velocity of rain as
experienced by her is the velocity of rain relative to the velocity of
the bicycle she is riding. That is vrb = vr – vb
• This relative velocity vector as shown in Fig. makes an angle θwith
the vertical. It is given by
Therefore, the woman should hold her umbrella at an angle of about 19°
with the vertical towards the west.

35. PROJECTILE MOTION
• Projectile is a body thrown with an initial velocity in the vertical plane
and then it moves in two dimensions under the action of gravity
alone without being propelled by any engine or fuel. Its motion is
called projectile motion. The path of a projectile is called its
trajectory.
• Projectile motion is a case of two-dimensional motion .
A body can be projected in two ways :
Horizontal projection- When the body is
given an initial velocity in the horizontal
direction only.
Angular projection- When the body is
thrown with an initial velocity at an angle to
the horizontal direction.

36. • ASSUMPTIONS:
□ There is no resistance due to air
□ Rotational motion of the earth is absent
□ Acceleration due to gravity is constant
both in magnitude and direction

37. HORIZONTAL PROJECTION
A body is thrown with an initial velocity u along the horizontal
direction. We will study the motion along x and y axis separately by
taking the starting point to be the origin.
Along x-axis
1. ux=u (Component along x-axis)
2. Acceleration along x-axis ax=0 (F =0)
3. Velocity in the horizontal direction
(here a=0),
4. The displacement along x-axis at any instant t,
Along y-axis
1. Component of initial velocity along y-axis. uy=0
2. Component of velocity along the y-axis at any instant t.
3. The displacement along y-axis at any instant t

38. EQUATION OF A TRAJECTORY
(PATH OF A PROJECTILE)
• We know at any instant x = ut
t=x/u
Also, y= (1/2)gt2
Substituting for t we get
y= (1/2)g(x/u)2
y= (1/2)(g/u2)x2
y= kx2 where k= g/(2u2 )
• Thus, the path of projectile, projected horizontally
from a height above the ground is a parabola.

39. TIME OF FLIGHT (T):
• It is the total time taken by a body to completes its projectile
motion.
• To find T , we will find the time for vertical fall
From y= u t + (1/2) gt2
y
Put y= h and time t = T
h= 0 + (1/2) gT2
T= (2h/g)1/2

40. RANGE (R) :
• It is the horizontal distance covered during the time
of flight T.
• From x= ut
When t=T , x=R
R=uT But T= (2h/g)1/2
R=u(2h/g)1/2 R

41. Net velocity & direction at any instant of
time t
• At any instant t
We know that
vx= u and vy= gt
x y
v= (v 2 + v 2)1/2
v= [u2 + (gt)2]1/2
• Direction of v with the horizontal at any instant :
y x
θ = tan-1 (v /v )= tan-1 (gt/u)

42. PROJECTILE FIRED AT AN ANGLE
WITH THE HORIZONTAL

43. Consider an object is projected with an initial velocity u at an angle
Φ to the horizontal direction.We will study the motion along x and y
axis separately by taking the starting point to be the origin.
Along X axis
Along Y axis

44. Equation of Trajectory
(Path of projectile)
This equation is of the form y= a x + bx2 where 'a'
and 'b are constants. This is the equation of a
parabola. Thus, the path of a projectile is a parabola
.

45. Time of flight (T)
• It is a time taken by a body to complete its projectile
motion.
• Time taken by projectile to reach the maximum height, Ta
(time of ascent)=T/2
Ta=Td (time of ascent =time of descent)
Also vy= usinΦ – gt
At t=T/2 , vy= 0 , So
0= u sinΦ – g (T/2)
T= (2usinΦ)/g

46. Maximum height (H)
MAXIMUM HEIGHT AT 900
Hmax= (u2)/2g

47. Horizontal Range (R) :
Range is the total horizontal distance covered during the
time of flight.
MAXIMUM RANGE AT 450
Rmax= u2/g

48. Net velocity of the body at any
instant of time t:
vx=ucosΦ
vy=usinΦ – gt
2 2 1/2
v= (vx + vy )
Φ= tan-1(v /v )
y x
Where Φ is the angle that the resultant velocity(v)
makes with the horizontal at any instant .
Note: projectile-motion_en.html

49. CIRCULAR MOTION
It is a movement of an object along the
circumference of a circle or rotation along
a circular path. It can be uniform, with
constant angular rate of rotation and constant
speed, or non-uniform with a changing rate of
rotation.

50. ANGULAR DISPLACEMENT
The angular displacement is defined as
the angle through which an object
moves on a circular path.
It is the angle, in radians,
between the initial and final positions.
(θ2- θ1) = θ angular displacement
θ = s/r
θ = angular displacement through
which movement has occurred
s = distance travelled
r = radius of the circle

51. ANGULAR VELOCITY
It is a quantitative expression
of the amount of rotation that
a spinning object undergoes
per unit time. It is a vector
quantity, consisting of an
angular speed component
and either of two defined
directions or senses.

52. TIME PERIOD:
Time taken by a particle to complete one
rotation is called time period. It is
denoted by T.
FREQUENCY:
Number of rotations completed by the
particle in one unit second is called
frequency. It is denoted by v.

53. RELATION BETWEEN ANGULAR
VELOCITY and TIME
Let dθ be the angular displacement made by the
particle in time dt , then the angular velocity of the
particle is dθ /dt
ω = dθ /dt .
Its unit is rad s-1
For one complete revolution, the angle swept by the
radius vector is 3600 or 2π radians.
then the angular velocity of the particle is
ω= θ/t = 2 π/T .
If the particle makes n revolutions per second,
then ω=2π(1/T) = 2πn
where n = 1/T is the frequency of revolution.

54. RELATION BETWEEN ANGULAR
VELOCITY AND TIME
Let PQ = ds, be the arc length covered by the particle
moving along the circle, then the angular displacement
d θ is expressed as dθ = ds/r.
But ds=vdt.

55. UNIFORM CIRCULAR MOTION
Uniform circular motion can be described as
the motion of an object in a circle at a constant
speed. As the object moves in a circular path, its
direction changes constantly. At all instances, the
direction of the object is tangent to the circle.

56. ANGULAR ACCELERATION
Angular acceleration is the rate of change of angular velocity.
In SI units, it is measured in radians per second squared
(rad/s2)
It is denoted by the Greek letter alpha (α).

57. RELATION BETWEEN ANGULAR
ACCELERATION AND LINEAR
ACCELERATION
Linear acceleration = Rate of change of linear velocity
a= dv/dt
But v=r ω
Therefore a= d(r ω)/ dt = r dω/ dt
Or a= r α
In vector form;

58. CENTRIPETAL ACCELERATION
When an object follows a circular path at a constant speed, the motion of the
object is called uniform circular motion. From fig
The acceleration is always directed
towards the centre. This is why this
acceleration is called centripetal
acceleration.