Acceleration

2) Derivation for the Equation of Motion s  = ut+at2
Velocity is defined as the rate of change of displacement. This is mathematically represented as:
    \( Velocity = \frac{{Displacement}} {{Time}} \)
Rearranging, we get
Displacement = Velocity x Time
If the velocity is not constant then in the above equation we can use average 
velocity in the place of velocity and rewrite the equation as follows:
    \( Displacement = \frac{{Initial Velocity + Final Velocity}} {2}x\,Time \)
           \( s = \frac{{v + u}} {2}\,x\,t \)
From the first equation of motion, we know that v = u + at. Putting this value of v
    in the above equation, we get
  \( s = \frac{{v + (u + at)}} {2}\,x\,t \) 
  \( s = \frac{{2u + at}} {2}\,x\,t \)
    \( s = \left( {\frac{{2U}} {2} + \,\frac{{at}} {2}} \right)\,x\,t \) 
 \( s = \left( {u + \,\frac{{at}} {2}} \right)x\,t \)  
On further simplification, the equation becomes: s  = ut+at2......(ii)