This is an update and expansion of the post here.
Here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time = t, location = x, initial location = x(t0) = x0, velocity = v, initial velocity = v(t0) = v0, speed = v = |v|, and acceleration = a.
First equation of motion
v = ∫ a dt = v0 + at
Second equation of motion
x = ∫ (v0 + at) dt = x0 + v0t + ½at²
Third equation of motion
From v² = v ∙ v = (v0 + at) ∙ (v0 + at) = v0² + 2t(a ∙ v0) + a²t², and
(2a) ∙ (x ‒ x0) = (2a) ∙ (v0t + ½at²) = 2t(a ∙ v0) + a²t² = v² ‒ v0², it follows that
v² = v0² + 2(a ∙ (x ‒ x0)), or
v² − v0² = 2a ∙ x, with x0 = 0.
Here is a derivation of the time-space equations of motion, in which retardation is constant. Let stance = x, time (chronation) = t, initial time = t(x0) = t0, lenticity = w, initial lenticity = w(x0) = w0, pace w = |w|, and retardation = b.
First equation of motion
w = ∫ b dx = w0 + bx
Second equation of motion
t = ∫ (w0 + bx) dx = t0 + w0x + ½bx²
Third equation of motion
From w² = w ∙ w = (w0 + bx) ∙ (w0 + bx) = w0² + 2x(b ∙ w0) + b²x², and
(2b) ∙ (t ‒ t0) = (2b) ∙ (w0x + ½bx²) = 2x(b ∙ w0) + b²x² = w² ‒ w0², it follows that
w² = w0² + 2(b ∙ (t ‒ t0)), or
w² − w0² = 2b ∙ t, with t0 = 0.
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