obtain equations of motion for constant …

Let uniform acceleration be a

Acceleration is defined as rate of change of velocity

  ......................(1)
where v is velocity of an object at time t

By integrating above ewqn.(1) , we get

where u is velocity at time t = 0 and v is velocity after time t seconds

from above integration we get ,  v - u = a t   or  v = u + ( a t )  ...................... (2)

velocity is defined as rate of change of displacement 

 ...................(3)

Where S is distance travelled.

By integrating above equation (2) , we get

Where x_o is initial position and x is position after time t

x - x_o = u t + (1/2) a t² .................. (4)

If the object travelled a distance S staring from initial velocity u and attained final velocity v after travelling distance S in time t , then we have

S = (1/2) ( u+v ) t  ...............(5)

Let us substitute t from eqn.(2) in eqn.(5) as  t = [ ( v - u ) / a ] , then we have

S = (1/2) ( u + v ) [ ( v - u ) /a ]    or  v² = u² + ( 2 a S )  ..................  (6)

Eqn.(2) , eqn.(4) and eqn.(6) are equations of motion