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Equations of Motion Generalized

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

(2a) ∙ (xx0) = (2a) ∙ (v0t + ½at²) = 2t(av0) + a²t² = v² ‒ v0², it follows that

v² = v0² + 2(a ∙ (xx0)), or

v² − v0² = 2ax, 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 = ww = (w0 + bx) ∙ (w0 + bx) = w0² + 2x(bw0) + b²x², and

(2b) ∙ (tt0) = (2b) ∙ (w0x + ½bx²) = 2x(bw0) + b²x² = w² ‒ w0², it follows that

w² = w0² + 2(b ∙ (tt0)), or

w² − w0² = 2bt, with t0 = 0.


Here is a derivation of the space-time 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 = ∫ F dt = p0 + Ft

Second weighted equation of motion

ξ = ∫ (p0 + Ft) dt = ξ0 + p0t + ½Ft²

Third weighted equation of motion

mv² − mv0² = 2max = 2Fx

½mv² − ½mv0² = max = Fx = ΔKE = W,

where KE is the kinetic energy and W is the work.


Here is a derivation of the time-space weighted equations of motion with weight n (vass), in which retardation is constant. Let stance = x, weighted time (chronation) = τ = nt, initial weighted chronation = τ(x0) = nt0, fulmentum = q = nw, initial fulmentum = q(x0) = q0 = nw0, scalar fulmentum q = |q|, and weighted retardation (release) = R = nb.

First weighted equation of motion

q = ∫ R dx = q0 + Rx

Second weighted equation of motion

τ = ∫ (q0 + Rx) dx = τ0 + q0x + ½Rx²

Third weighted equation of motion

nw² − nw0² = 2nbt = 2Rt

½nw² − ½nw0² = nbt = Rt = ΔKL = Z,

where KL is the kinetic lethargy and Z is the repose (inverse of work).

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