iSoul In the beginning is reality

# 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 = Δs/Δt u = Δt/Δs ω = Δθ/Δt = v/Rs ψ = Δφ/Δs = u/Rt Average Rate 2 a = Δv/Δt b = Δu/Δs α = Δω/Δt β = Δψ/Δs Instantaneous Rate Velocity v = ds/dt = 1/u Tempo u = dt/ds = 1/v Angular velocity ω = dθ/dt = dt/dφ Angular tempo ψ = dφ/ds = ds/dθ Instantaneous Rate 2 Acceleration a = dv/dt := 1/b Expedience b = du/ds := 1/a Tangential acceleration α = dω/dt Tangential expedience β = dψ/ds Centripetal Radial Rate 2 Centripetal acceleration acen = v2/Rs = v/Rt Centripetal expedience bcen = u2/Rt = u/Rs Radial acceleration arad = Rs ω2 Radial expedience 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 = (v ‒ v0)/a ω = ω0 + αt t = (ω ‒ ω0)/α Second Equation of Space-Time s = s0 + v0t + ½at² t = (-u0/a) + √[(u0/a)2 + 2(s ‒ s0)/a] θ = θ0 + ω0t + ½αt2 φ = (-β/ψ0) + √[(β/ψ0)2 + 2β(s ‒ s0)] Third Equation of Space-Time v² = v0² + 2a(s – s0) s = s0 + (v² ‒ v0²)/2a ω² = ω0² + 2α(θ – θ0) θ = θ0 + (ω2 ‒ ω02)/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(t ‒ t0)/b] t = t0 + u0s + ½bs² θ = (-α/ω0) + √[(α/ω0)2 + 2α(t ‒ t0)] φ = φ0 + ψ0t + ½βs2 Third Equation of Time-Space t = t0 + (u2 ‒ u02)/2b u² = u0² + 2b(t – t0) φ = φ0 + (ψ2 ‒ ψ02)/2β ψ² = ψ0² + 2β(φ – φ0)