Motion is a change in position of an object or else a process of moving or being moved. When the body changes its position with respect to its surrounding, the body is said to be in Motion.
Examples: Football on ground, motion of moon around earth, rock falling off a cliff, a car moving on the road to trees on the roadside, person inside a moving bus with respect to person outside the bus, bird flying in sky are the examples of motion.
The minimum distance between two points is called
displacement while the actual path covered is called distance. The
displacement is a vector term and distance is scalar term. Distance and
displacement both have SI unit as meter.
$AB + BC$ = distance moved and $AC$ = displacement
The effect of $AB + BC$ is same as effect of $AC$.
On one round trip, distance is $2(AB + BC)$ while the displacement = $AC + CA$ = $0$
Hence the distance is never zero while the displacement is zero in one round trip.
As we know that the rate of change of displacement is velocity similarly we have,
$Speed$ = $\frac{Distance\ moved}{Time\ taken}$
$S$ = $\frac{d}{t}$
where $d$ is distance moved.
The $SI$ unit for velocity and speed is meter/second $(m/s)$.
The speed is scalar term and velocity is vector term.
The speed cannot be zero since distance cannot be zero while the velocity can be zero as displacement can be zero.
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The types of motion are:
 Uniform motion
 Non uniform motion
a) Uniform motion: When equal distance is covered in equal interval of time, the motion is said to be in uniform motion. The bodies moving with constant speed or velocity have uniform motion or increase at the uniform rate.b) Non Uniform motion: When unequal distances are covered in equal interval of time, the motion is said to be in non uniform motion. The bodies executing non uniform motion have varying speed or velocity.We can even classify motion into three types:
 Translatory motion
 Rotatory motion
 Vibratory motion
Translatory Motion
In Translatory motion, the particle moves from one point in space to another. This motion may be along a straight line or along a curved path.
They can be classified as:
 Rectilinear Motion: Motion along a straight line is called rectilinear motion.
 Curvilinear Motion: Motion along a curved path is called curvilinear motion.
Rotatory Motion
In Rotatory motion, the particles of the body describe concentric circles about the axis of motion.
Vibratory Motion
In Vibratory motion, the particles move to and fro about a fixed point.
The variable quantities in a uniformly accelerated rectilinear motion
are time, speed, distance covered and acceleration. Simple relations
exist between these quantities. These relations are expressed in terms
of equations called equations of motion.
There are three equations of motion.
1) $v$ = $u + at$
2) $S$ = $ut +$ $\frac{1}{2}$ $at^2$
3) $v^2$ = $u^2 + 2as$
Where,
$v$ = Final velocity
$u$ = Initial velocity
$a$ = Acceleration
$s$ = Distance traveled by a body
$t$ = Time taken.
First Equation of Motion:
Consider a particle moving along a straight line with uniform acceleration '$a$'. At $t$ = $0$, let the particle be at $A$ and $u$ be its initial velocity and when $t$ = $t$, $V$ be its final velocity.
$Acceleration$ = $\frac{change\ in\ velocity}{Time}$
= $\frac{vu}{t}$
$at$ = $v  u$
$v$ = $u + at$ ........ First equation of motion.
Average Velocity = Average Velocity = .....(1) Average Velocity can be written as Average Velocity = ........(2) From equations (1) and (2) = .......(3) The first equation of motion is $v$ = $u + at$.Substituting the value of $v$ in equation (3) we get
Second Equation of Motion:
$\frac{Total\ distance\ traveled}{Total\ time\ taken}$
$\frac{s}{t}$
$\frac{u + v}{2}$
$\frac{u + v}{2}$
$\frac{s}{t}$ $\frac{u + v}{2}$
$\frac{s}{t}$ = $\frac{(u + u + at)}{2}$
$s$ = $\frac{(2u + at)t}{2}$ = $\frac{2ut + at^{2}}{2}$ = $\frac{2ut}{2}$ + $\frac{at^2}{2}$
Which gives the second equation of motion.
Third equation of Motion:
The first equation of motion is $v$ = $u + at$.
$v  u$ = $at$ ... (1)
Average velocity = $\frac{s}{t}$ ... (2)
Average velocity = $\frac{u + v}{2}$ ... (3)
From equation (2) and equation (3) we get,
$\frac{u + v}{2}$ = $\frac{s}{t}$ ... (4)
Multiplying eq (1) and eq (4) we get,
$(v  u)(v + u)$ = $at \times$ $\frac{2s}{t}$
$(v  u)(v + u)$ = $2as$
[We make use of the identity $a^2  b^2$ = $(a + b) (a  b)]$
$v^2  u^2$ = $2as$ .......................... Third equation of motion.
Derivation of Equation of Motion Graphically
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First Equation of Motion
Consider an object moving with a uniform velocity u in a straight line. Let it be, given a uniform acceleration at time, t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and s is the distance covered by the object in time t. The figure shows the velocitytime graph of the motion of the object.
Slope of the v  t graph gives the acceleration of the moving object.
Thus, acceleration = slope = AB = $\frac{BC}{AC}$ = $\frac{v  u}{t  0}$ a = $\frac{v  u}{t}$
v  u = at
v = u + at................................................................(1)
Second Equation of Motion
Let u be the initial velocity of an object and 'a' the acceleration produced in the body. The distance traveled s in time t is given by the area enclosed by the velocitytime graph for the time interval 0 to t.
Distance traveled s = area of the trapezium ABDO
= area of rectangle ACDO + area of Î”ABC
= OD x OA +
$\frac{1}{2}$ BC x AC
= t x u +
$\frac{1}{2}$ (v  u) x t
= ut +
$\frac{1}{2}$ (v  u) x t
(v = u + at I eqn of motion; v  u = at)
S = ut +
$\frac{1}{2}$at x t
S = ut + $\frac{1}{2}$at^{2}.
Third Equation of Motion
Let 'u' be the initial velocity of an object and a be the acceleration produced in the body. The distance travelled 's' in time 't' is given by the area enclosed by the v  t graph.
S = area of the trapezium OABD.
= $\frac{1}{2}$ (b_{1} + b_{2})h
= $\frac{1}{2}$ (OA + BD) AC
= $\frac{1}{2}$ (u + v)t ....(1)
But we know that a = $\frac{v  u}{t}$
Or t = $\frac{v  u}{a}$
Substituting the value of t in eq. (1) we get,
s =
$\frac{1}{2}$ $\frac{(u + v)(v  u)}{a}$ =
$\frac{1}{2}$ $\frac{(v + u)(v  u)}{a}$2as = (v + u)(v  u)
(v + u)(v  u) = 2as [using the identity a
^{2}  b
^{2} = (a + b) (a  b)]
v^{2}  u^{2} = 2as........... Third Equation of Motion
a) For body moving at constant
velocity:The graph of straight line parallel to the
X axis shows that the body is moving with constant velocity.
b) For uniform motion:This graph shows the equal displacement in equal interval of time so, the
slope =
$\frac {\Delta Y} {\Delta X}$ gives the change in position over corresponding change in time is constant. Thus, this graph shows the uniform motion.
c) For Body at Rest:The positiontime graph parallel to time axis shows that the body is at rest.
d) For Non uniform motion:This graph shows unequal distance in
equal interval of time which gives the change in
position over corresponding change in time which is varying.
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Motion can be angular or uniform. When
the body moves on a curved path, there is a change in angular
displacement, this is called an angular motion. The rate of
change of angular displacement gives angular velocity. It’s
a vector term. The angular motion is always an accelerated motion.
Angular velocity (ω) = $\frac {d \theta} {dt}$ Where
d$theta$ is angular displacement.
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When the body moves in straight path, equal change in linear displacement in equal interval of time gives uniform motion.
Uniform velocity (v) = $\frac{dS}{dt}$
Where, dS is change in linear displacement
and
dt is the time taken.
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Speed = $\frac{Distance\ moved}{Time\ taken}$
S = $\frac{d}{t}$
where, d is distance moved.
The SI unit for velocity and speed is meter/second (m/s). The speed is scalar term and velocity is vector term. The speed cannot be zero since distance cannot be zero while the velocity can be zero as displacement can be zero.
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Newton has given the three laws of motion.
Newton’s First law of motion: The body remains in rest or in continue
motion unless some external force is applied on it. Example: The book on
table will remain on table unless some force is applied on it. The ball
moving on ground stops by itself because of friction (external
force). If there were no frictional forces, the moving ball will
continue to move unless we stop it.
First law of motion is related to term “Inertia”. It’s the property of
body by the virtue of which the body resists the external force.
Common examples of inertia in our day to day life:
 The passengers fall forward when the bus suddenly stops. This is due to
inertia of motion, the lower portion of body comes to rest but the upper
portion of body continue to be in motion.

When we shake the branches, the fruits and leaves fall. The branches are
in motion while the fruits and leaves are in rest so, they fall.

The dust particles get removed when we shake the carpet. This is, because
the particles are at rest while the carpet is moving, so, the particles are removed.

When the person jumps from the moving bus, he runs through some distance due to inertia of motion.

Any moving body has momentum. Mathematically, the momentum is denoted by P. It’s the product of mass and its velocity.
P = mass x velocity
P = m x v
Newton’s Second law of motion: According to the Newton’s second law of
motion the rate of change of momentum is directly proportional to the force
applied and acts in the direction of force.
F = $\frac{dP}{dt}$ ……………………………….(1)
dP is change in momentum
F = d $\frac{(mv – mu)}{dt}$……………………(2)
m = mass of body.
F = m $\frac{d(v  u)} {dt}$
$\therefore$ F = ma …………………….(3)
a = $\frac{v – u}{t}$
Force = mass x acceleration
Hence, a = $\frac{F}{m}$
For constant force, acceleration produced in the body, is inversely proportional to the mass of the body. Larger is mass, lesser is acceleration produced.
For equal masses of the body, the acceleration is directly proportional to the force applied. Larger is force, higher is acceleration produced.
Newton’s Third law of motion: To every action, there is an equal and opposite reaction.Common examples of Newton’s third law in our day today life:
 When a person jumps from a boat, the boat moves backwards.

When a bullet is fired, the gun goes backwards.

The huge amount of smoke downward, pushes the rocket upwards.
 When a balloon is blown, the air rushes outward while the balloon moves backward with the same momentum.