Question

S=ut+1/2atsqure derive calculus method

Answer 1

s = displacement, v = velocity, a = acceleration.

we know that by definition:    v = ds/dt,     a = dv/dt
Initial conditions:   Let the velocity at t=0 be u.  Let s = 0 at t = 0.
Let  a be constant.  u is a constant.

                         a = dv/dt,    =>   dv = a dt
Integrate on both sides from t =0 to t = t,
                          v - u = a (t - 0) = at
                          So  v = u + a t
Integrate wrt t again,
                          int v dt = int u dt  + integral a t  dt
                          s - 0  =  u * (t - 0) +  a * (t² - 0²)/2
                           s = u t + 1/2 a t²