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Motion

When we observe our surroundings, we can see many physical interactions taking  place around us like a book falling, an ear drum vibrating, bus moving, nuclear reactions etc. Everything in the universe moves. It can either be a small amount of movement or swift, but movement does happen this change in position of an object is called Motion. 

If an object is moving, we would be curious to know what are the things happening that make a body move, how long will a body move and many other queries pop in. 

To understand the importance of Motion we have a clear example mentioned below:
Motion

An object tends to continue in its motion at a constant velocity until and unless an outside force acts on it. The term velocity refers both to the speed and the direction in which an object is moving. It is easy to recognize an object in motion and an object at rest. One must apply an external force to disrupt the balance. 

The following are the terms to be recognized before learning Motion: 

Rest: When the body does not change its position with respect to the surroundings, the body is said to be at rest. 

Motion and Rest are relative terms.
For example: The person sitting inside the moving train is at rest, whereas the person sitting next to him but who is at Motion with the person outside the moving train. 

A book on the table is at rest with respect to the table and other objects in the room. But all these objects are sharing the motion of the earth

Or

A car moving on a road is said to be in motion compared to the poles and trees on the roadside. But the people sitting inside the car are at rest compared to one another.

 

What is Motion?

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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.
Displacement vector
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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Types of Motion

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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:
  1. Rectilinear Motion
    Motion along a straight line is called rectilinear motion.
  2. 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.

Equations of Motion

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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 + $\frac{1}{2}$ at2
3) v2 = u2 + 2as

Where,
v = Final Velocity         
u = Initial velocity         
a = acceleration         
s = distance traveled by a body
t =  time taken.

Derivation of Equation of Motion


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.


First Equation of Motion

Acceleration = $\frac{change\ in\ velocity}{Time}$
                    =  $\frac{v-u}{t}$.
at = v-u
v = u+ at   ........ First equation of motion.
Second Equation of Motion:

Average Velocity = $\frac{Total\ distance\ traveled}{Total\ time\ taken}$
Average Velocity = $\frac{s}{t}$.....(1)

Average Velocity can be written as $\frac{u+v}{2}$
Average Velocity = $\frac{u+v}{2}$........(2)
From equations (1) and (2)
$\frac{s}{t}$ = $\frac{u+v}{2}$ .......(3)

The first equation of motion is v = u + at.

Substituting the value of v in equation (3) we get

$\frac{s}{t}$ = $\frac{(u + u + at)}{2}$

s = $\frac{(2u + at)t}{2}$ = $\frac{2ut+at^{2}}{2}$ = $\frac{2ut}{2}$ + $\frac{at^2}{2}$ 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 x $\frac{2s}{t}$

(v - u)(v + u) = 2as

[We make use of the identity a2 - b2 = (a + b) (a - b)]
v2 - u2 = 2as.......................... Third equation of motion.

Derivation of Equation of Motion(Graphically)


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 velocity-time graph of the motion of the object.Graphical Deriavtive of First Equation

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

Graphical Derivative of Second Equation
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 velocity-time graph for the time interval 0 to t.

Distance traveled s = area of the trapezium ABDO

                            = area of rectangle ACDO + area of ΔABC

                            = 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 + $\frac{1}{2}$at2.

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.


Graphical Derivative of Third Equation
S = area of the trapezium OABD.

= $\frac{1}{2}$ (b1 + b2)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 a2 - b2 = (a + b) (a - b)]

v2 - u2 = 2as........... Third Equation of Motion
a) For body moving at constant velocity:
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:
Body at Rest
The position-time graph parallel to time axis shows that the body is at rest.

d) For Non uniform motion:

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, is angular displacement. → Read More 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.
→ Read More
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's Law of Motion

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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:

  1. 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.
  2. 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.
  3. 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.
  4. When the person jumps from the moving bus, he runs through some distance due to inertia of motion.
  5. 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 = $\frac{dP}{dt}$ ……………………………….dP is change in momentum

F = d $\frac{(mv – mu)}{dt}$……………………m = mass of body.

F = m $\frac{d(v - u)} {dt}$

Therefore F = ma …………………….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:

  1. When a person jumps from a boat, the boat moves backwards.
  2. When a bullet is fired, the gun goes backwards.
  3. The huge amount of smoke downward, pushes the rocket upwards.
  4. When a balloon is blown, the air rushes outward while the balloon moves backward with the same momentum.
More topics in Motion
Vector Distance and Displacement
Uniform Motion Angular Motion
Rectilinear Motion Motion Graphs
Speed Velocity
Newton's Laws of Motion Acceleration
Kinematics Equations Free Fall
Projectile Motion
Related Topics
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