Kinematics(class)

Index
Rectilinear Motion
1) Introduction: rest and motion, Frame of reference , distance/displacement
2) Speed and velocity
3) Acceleration
4) Non uniform acceleration
5) Graphs: distance/displacement-time graph, speed/velocity-time graph,
acceleration-time graph
Projectile motion
1. Introduction to projectile
2. Oblique projectile
3. Horizontal projectile
4. Projectile on a moving platform
5. Projectile on an inclined plane
6. Elastic collision of projectile
Relative motion
1) Introduction
2) River boat problems, Wind aero plane problems , Rain umbrella problems
3) Velocity of approach/separation, condition for collision , minimum distance

Mechanics




 motionKinema
Word
Greek
Kinematics

Concept of a Point
Object/point mass/particle
When the size of the object is much less in comparison to the distance
covered by the object then the object is considered as a point object.
Dimensions of car are small as compared to the distance travelled, so we
can take it as a point object.

Concept of a Point
Object/point mass/particle
A BODY - A certain amount of matter limited in all directions and consequently having a
finite size, shape and occupying some definite space is called a body.
RIGID BODY - A body is said to be rigid if the distance between any pair of its constituent
particles remains unchanged.
PARTICLE - The smallest part of matter with zero dimension which can be described by its
mass and position is defined as a particle. Thus a particle has only a definite position, but no
dimension. In the problems we are going to discuss, we will consider a body to be a particle
for the sake of simplicity.

Rest :
An object is said to be at rest if it does not change its position w.r.t. its
surroundings with the passage of time.
Motion :
A body is said to be in motion if its position changes continuously w.r.t.
the surroundings (or with respect to an observer) with the passage of
time.
Motion and Rest

Motion and Rest
In everyday life, we see some objects at rest and others in motion.
Eg Birds fly, fish swim,blood flows through veins and arteries and cars move.
Atoms, molecules, planets, stars and galaxies are all in motion.
Fig: tom and jerry in motion(left) and tom and jerry at rest (right)

REST AND MOTION ARE RELATIVE TERMS
Motion is a combined property of the object and the observer. There is no
meaning of rest or motion without the observer. Nothing is in absolute rest or
in absolute motion
For the passengers in a moving bus, the roadside trees appear to be moving
backwards. A person standing on the road–side perceives the bus along with
the passengers as moving. However, a passenger inside the bus sees his
fellow passengers to be at rest.

There is nothing like absolute
motion or absolute rest
We know that the earth is rotating about its axis and revolving around the sun. The stationary
objects like your classroom, a tree and the lamp posts etc., do not change their position with
respect to each other i.e. they are at rest.
Although earth is in motion. To an observer situated outside the earth, say in a space ship, our
classroom, trees etc. would appear to be in motion. Therefore, all motions are relative. There is
nothing like absolute motion.

Frame of Reference
A passenger standing on platform observes that tree on a platform is at rest
But when the same passenger is passing away in a train through station, observes that tree is in
motion. In both conditions observer is right.
But observations are different because in first situation observer stands on a platform, which is
reference frame at rest and in second situation observer moving in train, which is reference
frame in motion.
It is a system to which a set of coordinates are attached and with reference to which observer
describes any event

Frame of Reference
Frame of reference in left image are people outside the bus, frame of reference in right
image are people inside the bus
• The person observing motion is known as observer.
• The observer for the purpose of investigation must have its own clock to measure
time and a point in the space attached with the other body as origin and a set of
coordinate axes.
• These two things the time measured by the clock and the coordinate system are
collectively known as reference frame

Frame of Reference and Reference
Point
Frame of reference is usually considered as origin and position of moving object
is given by (x,y,z) coordinate
To locate the position of object we need a frame of reference.
1) A convenient way to set up a frame of reference is to choose three mutually
perpendicular axis and name them x-y-z axis.
2) The coordinates (x, y, z) of the particle then specify the position of object w.r.t.
that frame.
3) If any one or more coordinates change with time, then we say that the object is
moving w.r.t. this frame.

TYPES OF MOTION:MOTIONS IN ONE,
TWO AND THREE DIMENSIONS
As position of the object may change with time due to change in one or two or all
the three coordinates, so we have classified motion as follows :
1) Motion in 1d
2) Motion in 2d
3) Motion in 3d

Types of frame of reference
Inertial Frame of Reference
When frame of reference(observer) is either at rest or in a state of constant
velocity, then those type of frame of references are known as inertial frames.
Non Inertial Frame of Reference
When frame of reference(observer) is in a state of accelerated motion, then
those type of frame of references are known as non inertial frames of
reference.
• For practical purposes, a frame of reference fixed to the earth can be considered
as an inertial frame . Strictly speaking, such a frame of reference is not an inertial
frame of reference, because the motion of earth around the sun is accelerated
motion due to its orbital and rotational motion.
• However, due to negligibly small effects of rotation and orbital motion, the
motion of earth may be assumed to be uniform and hence a frame of reference
fixed to it may be regarded as inertial frame of reference.

TYPES OF MOTION:MOTIONS IN ONE, TWO AND
THREE DIMENSIONS

Motion in 1-D
Eg. : (i) Motion of train along straight railway track.
(ii) An object falling freely under gravity.

Motion in 2-D
Eg. : (i) Motion of queen on carom board.
(ii) An insect crawling on the floor of the room.
(iii) Motion of object in horizontal and vertical circles etc.
(iv) Motion of planets around the sun.
(v) A car moving along a zigzag path on a level road.

Motion in 3-D
Eg.: (i) A bird flying in the sky (also kite).
(ii) Random motion of gas molecules.
(iii) Motion of an aero plane in space.

Types of motion
(i) Linear motion (or transalatory motion) : The motion of a moving car, a
person running, a stone being dropped.
(ii) Rotational motion : The motion of an electric fan, motion of earth about
its own axis.
(iii) Oscillatory motion : The motion of a simple pendulum, a body suspended
from a spring (also called to and fro motion).

Position Vector
POSITION VECTOR
It describes the instantaneous position of a particle with respect to the chosen frame of reference. It
is a vector joining the origin to the particle. If at any time, (x, y, z) be the cartesian coordinates of
the particle then its position vector is given by kˆzjˆyiˆxr 

.
In one dimensional motion :
(along x - axis) iˆxr 

, y = z = 0
In two dimensional motion : jˆyiˆxr 

, in x-y plane z = 0

r1

r2
y
O
r
B
A
x
A and B are position vectors

Displacement vector
DISPLACEMENT VECTOR
It is a vector joining the initial position of the particle to its final position after an interval of time.
Mathematically, it is equal to the change in position vector.
12Δ rrr



r1

r2
y
O
r
B
A
x
Line joining heads of both position Vectors is
displacement vector. Direction is given by subtraction of
position vectors

Position, Distance and
displacement are relative terms
Any object is situated at point O and three observers from three different places
are looking for same object, then all three observers will have different
observations about the position of point O and no one will be wrong. Because
they are observing the object from their different positions(origins).
Observer ‘A’ says : Point O is 3 m away in west direction.
Observer ‘B’ says : Point O is 4 m away in south direction.
Observer ‘C’ says : Point O is 5 m away in east direction.
Therefore position of any point is
completely expressed by two factors:
• Its distance from the observer and
• its direction with respect to observer.
That is why position is characterized by a
vector known as position vector.

Distance v/s Displacement
Distance Displacement
1. Distance is the length of the path actually
traveled by a body in any direction.
1. Displacement is the shortest distance
between the initial and the final positions of a
body in the direction of the point of the final
position.
2. Distance between two given points depends
upon the path chosen.
2. Displacement between two points is
measured by the straight path between the
points.
3. Distance is always positive. 3. Displacement may be positive as well as
negative and even zero.
4. Distance is scalar quantity. 4. Displacement is a vector quantity
5. Distance will never decrease 5. Displacement may decrease.

Note
(i) The magnitude of displacement is equal to minimum possible distance between two
positions. In general magnitude of displacement is not equal to distance. However, it
can be so if the motion is along a straight line without change in direction.
Distance  Displacement
(ii) For a moving particle distance can never be negative or zero while
displacement can be.(zero displacement means that body after motion has came
back to initial position)
i.e., Distance > 0 but Displacement > = or < 0
(iii) For motion between two points displacement is single valued while distance
depends on actual path and so can have many values.
(iv) For a moving particle distance can never decrease with time while
displacement can. Decrease in displacement with time means body is moving
towards the initial position.

Special cases: Displacement
Cases of Zero displacement
If a body travels in such a way that it comes back to its starting
position, then the displacement is zero. However, distance
traveled is never zero.
(i) When an object remains stationary or it moves first towards
right and then an equal distance towards left, its
displacement is zero.
(ii) Circular motion

Questions:
position,displacement,distance

Types of Motion: Uniform motion
• For example, a car running at a constant speed of say, 10 meters per second, will
cover equal distances of 10 meters every second, so its motion will be uniform
• A body has a uniform motion if it travels equal distances in equal intervals
of time, no matter how small these time intervals may be.
Examples of uniform motion –
(i) An aero plane flying at a speed of 600 km/h
(ii) A train running at a speed of 120 km/h
(iii) Light energy travelling at a speed of 3 × 108 m/s
(iv) A spaceship moving at a speed of 100 km/s

Types of motion: Non-Uniform
Motion
A body has a non-uniform if it travels unequal distances in equal intervals
of time. For example, if we drop a ball from the roof of a building, we will
find that it covers unequal distances in equal intervals of time.
Examples of non-uniform motion –
(i) An aeroplane running on a runway before taking off.
(ii) A freely falling stone under the action of gravity.
(iii) An object thrown vertically upward.
(iv) When the brakes are applied to a moving car.

Speedometer and Odometer
Speedometer measures instantaneous speed whereas odometer measures total distance travelled.

Speed
Different objects have different
speeds
Speed : Rate of change of distance covered per unit time is called speed.
(i) It is a scalar quantity having symbol  .
(ii) Dimension : [M0L1T–1]
(iii) Unit : metre/second (S.I.), cm/second (C.G.S.)

Types of speed :Average and
Instantaneous speed
Average Speed :
It is the total distance traveled by the object divided by the total time taken to
cover that distance.
Average speed =
takentimetotal
travelledcedistotal tan

Average and Instantaneous speed
Instantaneous Speed
The speed of an object at any particular instant of time or at particular point of its
path is called the instantaneous speed of the object. It is measured by speedometer
in an automobile.
Instantaneous speed is
different for different instants

Average speed cases
Time average speed
When particle moves with different uniform speed  1 ,  2 ,  3 ... etc in
different time intervals t1 , t2 , t3 ... Etc. respectively, its average speed over
the total time of journey is given as
When particle moves with speed v1 up to half time of its total motion and
in rest time it is moving with speed v2 then

Special cases
When particle moves the first half of a distance at a speed of v1 and second half
of the distance at speed v2 then
Distance averaged speed
When a particle describes different distances d1 , d2 , d3 ...... with different
time intervals t1 , t2 , t3 , ...... with speeds  1 ,  2 ,  3 respectively then the
speed of particle averaged over the total distance can be given as

Speed with Direction-Velocity
• It is the rate of change of displacement.
Therefore, velocity =
• S.I. unit of velocity is m/s.
• It is a vector quantity.
• Magnitude of the velocity is known as speed

time
ntdisplaceme
takentime
directiongivenaintravelledcetandis
Direction of velocity represents direction of motion of body.
OR
Sign of velocity represent the direction of motion of body.

Average and Instantaneous velocity
Average Velocity :
It is defined as the ratio of its total displacement to the total time interval in which
the displacement occurs.
Average velocity =
If x1 & x2 are the positions of an object at times t1 & t2 then,
timeTotal
ntdisplacemeTotal
t
x
t
xx
v 12
av








12 ttt 
Instantaneous Velocity :
The velocity of an object at any given instant of time at particular
point of its path is called its instantaneous velocity.
dt
xd
t
x
V t




  0lim

Graphs
Calculation of average
velocity

Comparison between instantaneous
speed and instantaneous velocity
(a) Instantaneous velocity is always tangential to the path followed by the particle.
When a stone is thrown from point O then at point of projection the instantaneous
velocity of stone is v1 , at point A the instantaneous velocity of stone is v2 ,
similarly at point B and C are v3 and v4 respectively.
Average velocity is the rate of change of
displacement vector between initial and
final position per unit time.
(b) A particle may have constant instantaneous speed but variable instantaneous
velocity.
Example : When a particle is performing uniform circular motion then for every
instant of its circular motion its speed remains constant but velocity changes at every
instant.

Comparison between instantaneous
speed and instantaneous velocity
(c) The magnitude of instantaneous velocity is equal to the instantaneous speed. But
magnitude of displacement is not equal to distance.
(d) If a particle is moving with constant velocity then its average velocity and
instantaneous velocity are always equal.
(e) If displacement is given as a function of time, then time derivative of
displacement will give instantaneous velocity.

Comparison between average
speed and average velocity
(a) Average speed is scalar while average velocity is a vector both having
same units (m/s) and dimensions [LT ] 1.
(b) Average speed or velocity depends on time interval over which it is defined.
(c) For a given time interval average velocity is single valued while average
speed can have many values depending on path followed.

Rate of Change of Velocity :
Acceleration
• Rate of change of velocity per unit time.
• The change may be either in magnitude or in direction or in both.
• It is a vector quantity.
• Its S.I. unit is m/sec2 and CGS unit is cm/sec2

Acceleration and retardation
(i) Positive acceleration :
Velocity and acceleration vector must be in same
direction.(for 1 d motion)
Or when angle between them is acute(for any
motion)
Eg. Free fall( Motion of freely falling bodies
because of acceleration due to gravity)
(ii) Negative acceleration
(retardation):
Velocity and acceleration vector must be in
opposite direction. (for 1 d motion) Or
when angle between them is obtuse(for any
motion)
Eg. A car slows down in front of a tree.

Average and Instantaneous
acceleration
Acceleration =
time
velocityinitialvelocityfinal
t
uv
time
velocityinchange 



Instantaneous acceleration
Average acceleration
second time derivative of displacement gives acceleration. If velocity is given as a function
of position, then by chain rule

Uniform and Non uniform
acceleration
(a) Uniform Acceleration (Uniformly Accelerated Motion):
A body is said to have uniform acceleration if magnitude and direction of
the acceleration remains constant during particle motion.
Eg. Motion of a freely falling body

Equations of Motion in scalar and
vector form

Important points
3) Distance increases when the dot product of displacement and velocity vector is >0
There is acute angle between the two vectors.
4) Speed increases when the dot product of acceleration and velocity vector is >0.
There is acute angle between the two vectors.

Motion of Body Under Gravity
The force of attraction of earth on bodies, is called force of gravity.. It is
represented by the symbol g.
For motion of a body under gravity, acceleration will be equal to “g”, where g
is the acceleration due to gravity. Its normal value is 9.8 m/s2 or 980 cm/s2
or 32 feet/s2 .

Important point : Average velocity for
Uniform acceleration

Important Points
1) For a moving body there is no relation between the direction of instantaneous
velocity and direction of acceleration.
2) Acceleration can be positive, zero or negative. Positive acceleration means
velocity increasing, zero acceleration means velocity is uniform, negative
acceleration (retardation) means velocity is decreasing.
3) A body is thrown vertically upwards. If air resistance is to be taken
into account, then the time of ascent is less than the time of descent. t2 > t1

Important Points
4) A particle is dropped vertically from rest from a height. The time taken by it to fall
through successive distance of 1m each will then be in the ratio of the difference in the
square roots of the integers i.e.
5) The distance covered in the nth sec,
So distance covered in I, II, III sec, etc., will be
in the ratio of 1 : 3 : 5, i.e., odd integers only.

Questions on uniform acceleration

Uniform and Non uniform
acceleration
b) Non-Uniform Acceleration :
A body is said to have non-uniform acceleration, if magnitude or direction or
both, change during motion.
Eg.1 Car moving in a crowded street.
Equations used in Non uniform acceleration.
When acceleration varies with time
When acceleration varies with distance

Kinematics In 3 Dimensions
Its position vector is defined by equations
Or we can analyze the motions in
3 different directions considering
them as 3 separate motions

Position and velocity in Non-
Uniform Acceleration

Questions on non uniform
acceleration

Distance/Displacement time graph
The change in the position of an object with time can be
represented on the distance-time graph adopting a convenient
scale of choice.
• The slope of the distance-time graph gives the speed of the
body.
The slope of the displacement-time graph gives the velocity of
the body.
• If slope of distance-time graph increases or decreases, the
speed of the body increases and decreases respectively.
Distance time graph and Displacement time graph are same
if body doesn’t change its direction.

Cases in Position time graph
Case 1: When the body is at rest
Case 2: When the body is in uniform speed
 = 00 so v = 0
i.e., line parallel to time axis
represents that the particle is at rest.
 = constant so v = constant, a = 0
i.e., line with constant slope represents uniform
velocity of the particle

Cases in Distance time graph
Case 3 :When the body is in motion with non-uniform (variable) increasing
speed (constant acceleration)
Case 4 :When the body is in motion with non-uniform (variable) decreasing
speed (constant acceleration)
 is increasing so v is increasing, a is positive.
i.e., line bending towards distance axis
represents increasing velocity of particle.
It means the particle possesses acceleration.
 is decreasing so v is decreasing, a is
negative
i.e., line bending towards time axis represents
decreasing velocity of the particle.
It means the particle possesses retardation.

Cases in Distance time graph
Case 5:When the body is in motion with constant speed but slope is negative
 constant but > 90o so v will be
constant but negative
i.e., line with negative slope represent
that particle returns towards the point
of reference. (negative displacement).

Conceptual question
Are these distance time graph possible with distance along y axes and time on x
axes?
No In first graph there are more than one positions for one time
And in second velocity can’t be infinite

Speed/Velocity time graph
The variation in velocity with time for an object moving in a straight line can be
represented by a velocity-time graph
• The slope of the velocity-time graph gives the velocity of the body.
• Area enclosed between the speed-time graph line and x-axis (time axis) gives the
distance covered by the body. Area enclosed between the velocity-time graph line
and the x-axis (time axis) gives the displacement of the body.

Cases of velocity/speed time graph
Case 1: When the body is at rest
Case 2: When the body is in uniform speed
 = 0, a = 0, v =0
i.e., line along time axis
represents that the particle is moving
with zero velocity.
 = 0, a = 0, v = constant
i.e., line parallel to time axis
represents that the particle is moving
with constant velocity.

Case 3 :When the body is in motion with non-uniform (variable)
increasing speed (constant acceleration)
Case 4 :When the body is in motion with non-uniform
(variable) decreasing speed(constant acceleration)
 =constant, so a = constant and v is
increasing uniformly with time
i.e., line with constant slope represents
uniform acceleration of the particle.
Negative constant acceleration because  is
constant and > 90obut initial velocity of
the particle is positive

Case 5:When the body is in motion with non-uniform (variable) acceleration
 increasing so acceleration increasing
i.e., line bending towards velocity axis
represent the increasing acceleration in the
body.
 decreasing so acceleration decreasing
i.e. line bending towards time axis
represents the decreasing acceleration in
the body

Projectile Motion
Any object that is given an initial velocity obliquely, and that subsequently follows a
path determined by the gravitational force (and no other force) acting on it, is called
a Projectile.
Examples of projectile motion :
• A cricket ball hit by the batsman for a six
• An aero plane dropping food packet/bomb
• A bullet fired from a gun.
Y
uy = u sin 
u

ux = u cos 
ux
R
H
X
Assumptions of Projectile Motion.
(1) There is no resistance due to
air.
(2) The effect due to curvature of
earth is negligible.
(3) The effect due to rotation of
earth is negligible.
(4) For all points of the
trajectory, the acceleration due to
gravity ‘g’ is constant in
magnitude and direction.

Types of Projectile
Projectile motion on an
inclined planeOblique projectile motion
Horizontal projectile motion
Any projectile motion can be broken down to two simultaneous motions in any two
perpendicular directions(say X and Y). These two motions are completely
independent from each other . This is known as Principle of Physical Independence
of Motions.

Important definitions
Range(R)
The point from where it is projected is known as point of projection, the point where it
falls on the ground is known as point of landing or target. The distance between these
two points is called range.
Maximum height(H)
The height from the ground of the highest point it reaches during flight is known as
maximum height.
Time of flight(T)
The duration for which it remain in the air is known as air time or time of flight.
Velocity and angle of projection(u and )
The velocity with which it is thrown is known as velocity of projection.
The angle which velocity of projection makes with the horizontal is known as angle of
projection.
Y
uy = u sin 
u

ux = u cos 
ux
R
H
X

PROJECTILE THROWN AT AN
ANGLE WITH HORIZONTAL
u

If a particle is projected from point O, at an angle  from the horizontal, with initial velocity u

then
the components of u

in x and y directions are given as
ux = u cos 
uy = u sin 
Y
uy = u sin 
u

ux = u cos 
ux
R
H
X
The horizontal component remains
unchanged throughout the flight. The force
of gravity continuously affects the vertical
component.
The horizontal motion is a uniform motion
and the vertical motion is a uniformly
accelerated motion

Y
uy = u sin 
u

ux = u cos 
ux
R
H
X
jsinuicosuu
juiuu yx



The X axis is parallel to the horizontal. Y axis is parallel to the vertical and the u lies in the plane X -
Y. The constant acceleration a is given as
jaiaa yx 
Where ax = 0 [ as there is no acceleration along the X - axis].
ay = -g [the acceleration is downward and equal to g].
Now velocity after time t is given as.
vtx = ux + axt = u cos (as ax = 0)
vty = uy + ayt = usin - gt
 jˆviˆvv yx 

   jˆ)gtsinu(iˆcosuv t 

The direction of v

with the x axis is given by 







x
y1
v
v
tan
Velocity of projectile

Displacement of projectile
  jˆ)gtsinu(iˆcosuv t 

(integrating the equation)

Equation of trajectory
Co-ordinates of the projectile after time t is given by
x = xo + uxt +
2
1
axt2
 x = 0 + u cos .t+ 0
 x = u cos t (1)
And y = yo + uyt +
2
1
ayt2
 y = 0 + u sin t -
2
1
gt2
 y = u sin t -
2
1
gt2
(2)
From equation (1) and (2)
We get,
)3(
cosu2
gx
tan.xy`
cosu
x
g
2
1
cosu
x
sinuy
22
2
22
2







Equation of trajectory
The above equation shows the relation between x and y and represents the path of the projectile
known as trajectory. The inspection of eq. (3) shows that u is the equation of parabola of the form
y = bx + cx2
Where b = tan  = a constant, and constanta
cosu2
g
c 22



Thus, the trajectory of a projectile is a parabola.

Time of Flight
Y
uy = u sin 
u

ux = u cos 
ux
R
H
X
Maximum Height
Horizontal Range

Special cases
i)
We get the same range for two angle of projections  and (90 – ) but in both
cases, maximum heights attained by the particles are different.
ii)

Horizontal Projectile
A body be projected horizontally from a
certain height ‘y’ vertically above the ground
with initial velocity u.
If friction is considered to be absent, then there
is no other horizontal force which can affect
the horizontal motion.
The horizontal velocity therefore remains
constant and so the object covers equal
distance in horizontal direction in equal
intervals of time

Displacement of Projectile (r )
x  ut
Instantaneous velocity

Time of flight
Horizontal range

Horizontal projectile
Case (ii) : Projection at an angle θ above horizontal
Case (i) : Horizontal projection
uy = usinθ , ay = – g
Case (iii) : Projection at an angle θ below horizontal

Variation of horizontal range with
vertical height

Variation of horizontal range with
projection speed

Projectile from moving platform
CASE (1) : When a ball is thrown upward from a truck moving with uniform speed,
then observer A standing in the truck, will see the ball moving in straight vertical line
(upward & downward).
CASE (2) : When a ball is thrown at some angle ‘θ’ in the direction of motion of the
truck, horizontal
ux = ucosθ + v and uy=usinθ
Horizontal & vertical component of ball’s velocity
w.r.t. observer B sitting on the ground, is
Horizontal & vertical component of ball’s velocity w.r.t.
observer B sitting on the ground, is
ux = u and uy=v

CASE (3) : When a ball is thrown at some angle ‘θ’ in the opposite direction of motion
of the truck, horizontal & vertical component of ball’s velocity w.r.t. observer A standing
on the truck, is u cosθ, and u sinθ respectively.
ux = ucosθ - v and uy=usinθ
CASE (4) : When a ball is thrown at some angle ‘θ’ from a platform moving with
speed v upwards, horizontal & vertical component of ball’s velocity w.r.t. observer A
standing on the moving platform, is ucosθ and usinθ respectively.
ux = ucosθ and uy=usinθ +v

CASE (5) : When a ball is thrown at some angle ‘θ’ from a platform moving with
speed v downwards, horizontal & vertical component of ball’s velocity w.r.t.
observer A standing on the moving platform, is ucosθ and usinθ respectively.
Horizontal & vertical component of ball’s velocity w.r.t. observer B sitting on the
ground, is
ux = ucosθ and uy=usinθ -v

Projectile on Inclined plane
Artillery application often finds target either up a hill or down a hill. These situations can
approximately be modeled as projectile motion up or down an inclined plane.

Case (i) : Particle is projected up the incline
Here α is angle of projection w.r.t. the inclined plane. x and y axis are taken
along and perpendicular to the incline as shown in the diagram.

Time of flight (T)
Maximum height (H)
Horizontal range(R)

Case (ii) : Particle is projected down the incline
Here α is angle of projection w.r.t. the inclined plane. x and y axis are taken
along and perpendicular to the incline as shown in the diagram.

Time of flight (T)
Maximum height (H)
Range along the inclined plane (R):

Rmax =
 β1
2
0
sing
v

up the plane and Rmax =
 β1
2
0
sing
v

down the pla
Maximum Range along the inclined plane (R):
For a given speed, the direction which gives the maximum range of the projectile on an
incline, bisects the angle between the incline and the vertical, for upward or downward
projection.
This happens for =45o-/2
This happens for =45o +/2
Up the inclined plane
Down the inclined plane
ne and Rmax =
 β1
2
0
sing
v

down the plane.

Elastic collision of a projectile with
a wall
Due to collision, direction of x component of velocity is reversed but its magnitude
remains the same and y component of velocity remains unchanged.
Therefore the remaining distance (R – x) is covered in the backward direction and the
projectile lands at a distance of R – x from the wall
Also time of flight and maximum height depends only on y component of velocity, hence
they do not change despite collision with the vertical, smooth and elastic wall.

Relative motion
Motion is a combined property of the object under study as well as the observer. It is
always relative ; there is no such thing like absolute motion or absolute rest. Motion is
always defined with respect to an observer or reference frame.
Reference frame :
Reference frame is an axis system from which motion is observed along with a clock
attached to the axis, to measure time. Reference frame can be stationary or moving.
Relative Position
ABAB rrr

/
ABr /
 is the position vector of B with respect to A
It is the position of a particle w.r.t. observer.

Relative motion
Relative Velocity
ABAB rrr

/
is the velocity of B with respect to A
Differentiating this
ABAB vvv

/
ABv /

Relative Acceleration
     ABAB r
dt
d
r
dt
d
r
dt
d 
/
ABAB vvv

/
Differentiating this
ABAB aaa /
Note: All velocities are relative & have no
significance unless observer is specified.
However, when we say “velocity of A”, what
we mean is , velocity of A w.r.t. ground which is
assumed to be at rest.
It is the rate at which relative velocity is
changing.

River boat problem
boat flowing
in direction of river
boat flowing Opposite to
direction of river
boat flowing at an angle to
direction of river
acute
obtuse
perpendicular

River boat
Case of minimum time to cross the river
Case of minimum drift to cross the river

Questions on river boat, aeroplane
wind, rain umbrella

Velocity of approach/separation
It is the component of relative velocity of one particle w.r.t. another, along the line
joining them.
If the separation is decreasing, we say it is velocity of approach and if separation is
increasing, then we say it is velocity of separation.
In one dimension, since relative velocity is along the line joining A and B, hence
velocity of approach / separation is simply equal to magnitude of relative velocity of A
w.r.t. B.
||/ ABseparationapproach vvv


Condition for uniformly moving particles to collide
If two particles are moving with uniform velocities and the relative velocity of one
particle w.r.t. other particle is directed towards the line joining each other then they
will collide.
Minimum / Maximum distance between two particles
If the separation between two particles decreases and after some time it starts
increasing then the separation between them will be minimum at the instant,
velocity of approach changes to velocity of separation. (at this instant v app = 0)

Questions on Velocity of
Approach / Separation

Kinematics(class)

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