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Motion equations revised

Previous posts on motion equations, for example here, showed a misleading parallelism. Although there is a formal parallelism as shown, it is more accurate to show the inverse equations. The parallel equations above and below have been revised accordingly.

Parallel Equations

  Linear w/3D space Linear w/3D time Angular w/3D space Angular w/3D time
Average Rate v = Δst u = Δts ω = Δθt = v/Rs ψ = Δφs = u/Rt
Average Rate 2 a = Δvt b = Δus α = Δωt β = Δψs
Instantaneous Rate Velocity

v = ds/dt = 1/u

Celerity

u = dt/ds = 1/v

Angular velocity

ω = dθ/dt = dt/dφ

Angular celerity

ψ = dφ/ds = ds/dθ

Instantaneous Rate 2 Acceleration

a = dv/dt := 1/b

Prestination

b = du/ds := 1/a

Tangential acceleration

α = dω/dt

Tangential prestination

β = dψ/ds

Centripetal

Radial Rate 2

Centripetal acceleration

acen = v2/Rs = v/Rt

Centripetal prestination

bcen = u2/Rt = u/Rs

Radial acceleration

arad = Rs ω2

Radial prestination

brad = Rt ψ2

Uniform Transverse Rate v = 2πRs/T u = 2πRt/S vtan = Rs ω utan = Rt ψ
Radius Spatial radius

Rs = S/(2π) = Rtv

Temporal radius

Rt = T/(2π) = Rsu

Spatial radius

Rs = ds/dθ = s/θ = v/ω

Temporal radius

Rt = dt/dφ = t/φ = u/ψ

Circumference

Arc Length

Circumference

S = 2πRs = 2πRtv

Circumference

S = 2πRt/u = 2πRs

Spatial arc length

θ = s/Rs

Temporal arc length

φ = t/Rt

Period T = 2πRs/v = 2πRt T = 2πRt = 2πRsu T = 2π/ω S = 2π/ψ
Position Distance: s Duration: t Arc distance: s = Rs θ Arc duration: t = Rt φ
Displacement s = s0 + vt t = (s ‒ s0)u θ = θ0 + ωt t = (θ θ0)ψRt2
First Equation of Space-Time v = v0 + at t = (vv0)/a ω = ω0 + αt t = (ωω0)/α
Second Equation of Space-Time s = s0 + v0t + ½at² t = (-u0/a) +

√[(u0/a)2 + 2(ss0)/a]

θ = θ0 + ω0t + ½αt2 φ = (-β/ψ0) +

√[(β/ψ0)2 + 2β(ss0)]

Third Equation of Space-Time = v0² + 2a(s s0) s = s0 + (v² ‒ v0²)/2a ω² = ω0² + 2α(θ θ0) θ = θ0 + (ω2ω02)/2α
Distimement s = (t ‒ t0)v t = t0 + us s = (φ φ0)ωRs2 φ = φ0 + ψs
First Equation of Time-Space 1/v = (1/v0) + (s/a) u = u0 + bs s =  (ψ ‒ ψ0)/β ψ = ψ0 + βs
Second Equation of Time-Space s = (-u0/b) +

√[( u0/b)2 + 2(tt0)/b]

t = t0 + u0s + ½bs² θ = (-α/ω0) +

√[(α/ω0)2 + 2α(tt0)]

φ = φ0 + ψ0t + ½βs2
Third Equation of Time-Space t = t0 + (u2u02)/2b u² = u0² + 2b(t t0) φ = φ0 + (ψ2ψ02)/2β ψ² = ψ0² + 2β(φ φ0)

 

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