Assumptions: Constant net force and constant acceleration

from kinematics:

d = 1/2 at² + v_ot     and     v = at + v_o

but  v_o = 0 , so

d = 1/2 at²             and      v = at

 

from Newton's second Law:

F = ma

a = F/m

 

substitution into the kinematic equations yields:

d = 1/2 (F/m) t²    and   

v = (F/m) t

 

solving for time gives:

t = v (m/F)

 

substituting into the displacement equation gives:

d = 1/2 (F/m) (v m/F)²

   = 1/2 (F/m) v²  (m² / F²)

   = 1/2 v² m / F

d F = 1/2 m v²

However, displacement x Force = Work, so:

W = 1/2 m v²

 

but by definition W = K so

 K = 1/2 m v²