If and object starts from rest and accelerates with a uniform or constant acceleration 'a' (increase in velocity rate per second) then after 't' seconds the object will have a velocity 'v' given by: -
v = at
Since the velocity is constantly changing over the 't' seconds from v=0 at t=0 to v=v at t=t we cannot simply multiply velocity by time to find the distance 's' covered. Instead we must use a mathematical method invented by Isaac Newton. The velocity sum 'at' must be integrated over infinitesimal elements of 'dt' for the period t=0 to t=t: -
Hence
...... t=t
s = ∫ a.t.dt
..... t=0
Thus, the result is the equation for distance 's' due to uniform acceleration 'a' over a period 't': -
s= ½at²