**Equations of Motion**

The equations of motion are mathematical expressions that describe the motion of an object in terms of its displacement, velocity, acceleration, and time. These equations are derived from Newton's laws of motion and are fundamental in analyzing and solving problems related to motion. There are three equations of motion that are commonly used to solve such problems, and they can be derived from the basic definitions of velocity and acceleration.

**First Equation of Motion: v = u + at**

This equation relates the final velocity (v) of an object to its initial velocity (u), acceleration (a), and time (t) taken to reach that velocity. It can be derived by rearranging the equation v = u + at such that:

v - u = at

This equation is applicable when the initial velocity, acceleration, and time are known, and we need to find the final velocity.

**Second Equation of Motion: s = ut + (1/2)at²**

This equation relates the displacement (s) of an object to its initial velocity (u), time (t), and acceleration (a). It can be derived by rearranging the equation s = ut + (1/2)at² such that:

s - ut = (1/2)at²

This equation is applicable when the initial velocity, acceleration, and time are known, and we need to find the displacement.

**Third Equation of Motion: v² = u² + 2as**

This equation relates the final velocity (v) of an object to its initial velocity (u), acceleration (a), and displacement (s). It can be derived by substituting the value of time (t) from the second equation of motion into the first equation of motion. The derivation is as follows:

v = u + at
t = (v - u) / a

Substituting the value of t in the second equation of motion:
s = ut + (1/2)at²
s = u((v - u) / a) + (1/2)a((v - u) / a)²
s = u(v - u)/a + (1/2)(v - u)²/a
s = (vu - u²)/a + (1/2)(v² - 2uv + u²)/a
s = (vu - u² + (1/2)v² - uv + (1/2)u²)/a
s = (1/2)v² - (1/2)u²

This equation is applicable when the initial velocity, acceleration, and displacement are known, and we need to find the final velocity.

In summary, the three equations of motion provide a mathematical framework to solve problems related to the motion of objects. By knowing the values of certain variables such as displacement, velocity, acceleration, and time, we can use these equations to calculate other unknown variables and analyze the motion of the object. Remember to use the appropriate equation based on the given information and rearrange it to isolate the desired variable.

1 equation:-
Acceleration= change in velocity / time
A= BC-DC / OC
A= BD / OC
= v-u / t
=> v - u = a × t
=> v - u = at
=> [ v = u + at ]

2 equation:-
s = ut + 1/2 at^2
s = Area of rectangle OADC+ Area of triangle ABD
s =(OA × OC )+ 1/2× AD × BD
s = ( u × t )+1/2× t (v - u)
s = ut + 1/2 ( v - u )× t
s = it +1/2 ( u - at - u ) t
s = [ut + 1/2 at ^2]

3 equation:-
v^2= u^2 + 2 a s
s= Area of trapezium OABC
s= 1/2 ( OA + BC ) OC
s= 1/2(u+v) (v-u /a)
s= 1/2(v+u)(v-u/a)
s= 1/2a (v+u)(v-u)
s= 1/2a (v^2 - u^2)
2as= v^2- u^2
v^2 - u^2= 2as
[ v^2= u^2+ 2as ]

That's it.....
I hope it will help you..