Recall the definitions of
• average velocity:
v[ave] = ∆x/∆t = (x[final] - x[initial])/t
Take the initial position to be the origin, so x[initial] = 0, and we simply write x[final] = s. So
v[ave] = s/t
• average acceleration:
a[ave] = ∆v/∆t = (v[final] - v[initial])/t
Assume acceleration is constant (a[ave] = a). Let v[initial] = u and v[final] = v, so that
a = (v - u)/t
Under constant acceleration, the average velocity is also given by
v[ave] = (v[final] + v[initial])/2 = (v + u)/2
Then
v[ave] = s/t = (v + u)/2 ⇒ s = (v + u) t/2
and
a = (v - u)/t ⇒ v = u + at
so that
s = ((u + at) + u) t/2
s = (2u + at) t/2
s = ut + 1/2 at²