Motion

Examples: Football on ground, motion of moon around earth, person inside a moving bus with respect to person outside the bus, bird flying in sky are the examples of motion.

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.
→ Read More The types of motion are:

• Uniform motion
• Non uniform motion
  

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 equation 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
and    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.

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.

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 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.



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:

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 position-time 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.

→ Read More

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, dθ 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 = $\frac{Distance\ moved}{Time\ taken}$.
→ Read More Newton has given the three laws of motion.

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 = v – u

Force = mass x acceleration

Hence, a = $\frac{F}{m}$