First, here is a derivation of the spatio-temporal equations of motion, in which acceleration is constant. Let time = t, location = s, initial location = s(t0) = s0, velocity = v, initial velocity = v(t0) = v0, v = |v| = speed, and acceleration = a.
First equation of motion
v = ∫ a dt = v0 + at
Second equation of motion
s = ∫ (v0 + at) dt = s0 + v0t + ½at²
Third equation of motion
From v² = v ∙ v = (v0 + at) ∙ (v0 + at) = v0² + 2t(a ∙ v0) + a²t², and (2a) ∙ (s ‒ s0) = (2a) ∙ (v0t + ½at²) = 2t(a ∙ v0) + a²t² = v² ‒ v0², it follows that
v² = v0² + 2(a ∙ (s ‒ s0)).
Next, here is a derivation of the temporo-spatial equations of motion, in which relentation is constant. Let base = s, chronation = t, initial chronation = t(r0) = t0, lenticity = w, initial lenticity = w(r0) = w0, w = |w| = pace, and relentation = b.
First equation of motion
w = ∫ b ds = w0 + bs
Second equation of motion
t = ∫ (w0 + bs) ds = t0 + w0s + ½bs²
Third equation of motion
From w² = w ∙ w = (w0 + bs) ∙ (w0 + bs) = w0² + 2s(b ∙ w0) + b²s², and (2b) ∙ (t ‒ t0) = (2b) ∙ (w0s + ½bs²) = 2s(b ∙ w0) + b²s² = w² ‒ w0², it follows that
w² = w0² + 2(b ∙ (t ‒ t0)).