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Equations of motion in space-time and time-space

First, here is a derivation of the space-time equations of motion, in which acceleration is constant. Let time (distimement) = t, position = r, initial position = r(t0) = r0, 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

r = ∫ (v0 + at) dt = r0 + v0t + ½at²

Third equation of motion

From v² = vv = (v0 + at) ∙ (v0 + at) = v0² + 2t(av0) + a²t², and

(2a) ∙ (rr0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (rr0)).


Next, here is a derivation of the time-space equations of motion, in which modulation is constant. Let position (displacement) = r, time = t, initial time = t(r0) = t0, allegrity = u, initial allegrity = u(r0) = u0, u = |u| = pace, and modulation = b.

First equation of motion

u = ∫ b dr = u0 + bt

Second equation of motion

t = ∫ (u0 + br) dr = t0 + u0r + ½br²

Third equation of motion

From u² = uu = (u0 + br) ∙ (u0 + br) = u0² + 2r(bu0) + b²r², and

(2b) ∙ (tt0) = (2b) ∙ (u0r + ½br²) = 2r(bu0) + b²r² = u² ‒ u0², it follows that

u² = u0² + 2(b ∙ (tt0)).

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