This is an update and expansion of the post here.
Here is a derivation of the chorocosmic 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 chronocosmic equations of motion, in which relentation is constant. Let stance = x, time (chronation) = t, initial time = t(x0) = t0, lenticity = w, initial lenticity = w(x0) = w0, pace w = |w|, and relentation = 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.
Here is a derivation of the chorocosmic weighted equations of motion with weight m (mass), in which acceleration is constant. Let time = t, weighted location = ξ = mx, initial weighted location = ξ(t0) = mx0, momentum = p = mv, initial momentum = p(t0) = p0 = mv0, scalar momentum p = |p|, and weighted acceleration (force) = F = ma.
First weighted equation of motion
p = ∫ ma dt = ∫ F dt = p0 + Ft
Second weighted equation of motion
ξ = ∫ (p0 + Ft) dt = ξ0 + p0t + ½Ft²
Third weighted equation of motion
mv² − mv0² = 2ma ∙ x = 2F ∙ x
½mv² − ½mv0² = ma ∙ x = F ∙ x = ΔKE = W,
where KE is the kinetic energy and W is the work.
Here is a derivation of the chronocosmic weighted equations of motion with weight n (vass), in which relentation is constant. Let stance = x, weighted time (chronation) = ζ = nt, initial weighted chronation = ζ(x0) = nt0, levamentum = q = nw, initial levamentum = q(x0) = q0 = nw0, scalar levamentum q = |q|, and weighted relentation (release) = R = nb.
First weighted equation of motion
q = ∫ nb dx = ∫ R dx = q0 + Rx
Second weighted equation of motion
ζ = ∫ (q0 + Rx) dx = ζ0 + q0x + ½Rx²
Third weighted equation of motion
nw² − nw0² = 2nb ∙ t = 2R ∙ t
½nw² − ½nw0² = nb ∙ t = R ∙ t = ΔKL = Z,
where KL is the kinetic lethargy and Z is the repose (inverse of work).