Mr. Rogers' AP Physics C: Mechanics - Derivation of K = 1/2 mv2
Definition of kinetic energy: the work done to accelerate an object from rest to its current velocity.

Assumptions: Constant net force and constant acceleration

from kinematics:

d = 1/2 at2 + vot     and     v = at + vo

but  vo = 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

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