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Equations of motion in time-space

The physics of time-space parallels that of space-time. Here is a derivation of the corresponding equations of motion, paralleling the exposition at the Physics Hypertextbook.

The one-dimensional equations of motion for constant prestination:

Let t = time, t0 = initial time, r = displacement, u = celerity, u0 = initial celerity, b = prestination.

First equation of motion

b = du / dr

du = b dr

∫ du from u0 to u = ∫ b dr from 0 to r

uu0 = br

u = u0 + br

Second equation of motion

u = dt / dr

dt = u dr = (u0 + br) dr

∫ dt from t0 to t = ∫ (u0 + br) dr from 0 to r

tt0 = u0r + ½br²

t = t0 + u0r + ½br²

Third equation of motion

du / dt = (du / dt) (dr / dr) = (du / dr) (dr / dt) = b / u

u du = b dt

u du from u0 to u = ∫ b dt from t0 to t

½(u² – u0²) = b(tt0)

u² = u0² + 2b(tt0)

Since displacement is in the denominator and time is in the numerator of celerity and prestination, these equations may be generalized to three dimensions of time.

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