Can you tell all derivation of motion

Dear Student,
Let,
Acceleration = a
Displacement = s
Initial velocity = u 
Final velocity = v
Time taken = t 

The first equation of motion.
a = change in velocity / t
  = v - u / t
Therefore ,
at = v - u
and, v = u + at 

Second equation of motion 

s = (average velocity ) t 
= ( u + v / 2 ) t  
=  (  u + u +at /2 ) t                    [ v = u + at]
= ( 2 u + at / 2 ) t
= ( 2u/2 + at/2 ) / t 
= ( u + at/ 2 ) t
= ut + 1/2 at2
Therefore,
s = ut + 1/2 at2

Third equation of motion 
v = u + at 
Therefore,
t = v - u / a............. (i)
s = ut + 1/2 at2
 = u ( v - u /a)    +  a/2 ( v - u / a )2            ...[ from (i)] 
Multiplying 2 on both sides of the equation , we get 
2 * s = 2 * [u  ( v - u ) a   +  a/2 ( v - u / a )]
2s = ‚Äč 2u ( v - u )/a    +  a ( v - u / a )2 
    = ( 2uv - 2u2)/ a +   a ( v2 + u- 2uv/ a2)
   =  ( 2uv - 2u2)/a  +  v2 + u- 2uv / a
 = ( 2uv - 2u2 + v2 + u2 - 2uv) / a
Simplifying , we get 
2as = v2- u2

Regards

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