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 |
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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/R_{s} | ψ = Δφ/Δs = u/R_{t} |
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 |
Modulation
b = du/ds := 1/a |
Tangential acceleration
α = dω/dt |
Tangential modulation
β = dψ/ds |
Centripetal
Radial Rate 2 |
Centripetal acceleration
a_{cen} = v^{2}/R_{s} = v/R_{t} |
Centripetal modulation
b_{cen} = u^{2}/R_{t} = u/R_{s} |
Radial acceleration
a_{rad} = R_{s} ω^{2} |
Radial modulation
b_{rad} = R_{t} ψ^{2} |
Uniform Transverse Rate | v = 2πR_{s}/T | u = 2πR_{t}/S | v_{tan} = R_{s} ω | u_{tan} = R_{t} ψ |
Radius | Spatial radius
R_{s} = S/(2π) = R_{t}v |
Temporal radius
R_{t} = T/(2π) = R_{s}u |
Spatial radius
R_{s} = ds/dθ = s/θ = v/ω |
Temporal radius
R_{t} = dt/dφ = t/φ = u/ψ |
Circumference
Arc Length |
Circumference
S = 2πR_{s} = 2πR_{t}v |
Circumference
S = 2πR_{t}/u = 2πR_{s} |
Spatial arc length
θ = s/R_{s} |
Temporal arc length
φ = t/R_{t} |
Period | T = 2πR_{s}/v = 2πR_{t} | T = 2πR_{t} = 2πR_{s}u | T = 2π/ω | S = 2π/ψ |
Position | Distance: s | Duration: t | Arc distance: s = R_{s} θ | Arc duration: t = R_{t} φ |
Displacement | s = s_{0} + vt | t = (s ‒ s_{0})u | θ = θ_{0} + ωt | t = (θ ‒ θ_{0})ψR_{t}^{2} |
First Equation of Space-Time | v = v_{0} + at | t = (v ‒ v_{0})/a | ω = ω_{0} + αt | t = (ω ‒ ω_{0})/α |
Second Equation of Space-Time | s = s_{0} + v_{0}t + ½at² | t = (-u_{0}/a) +
√[(u_{0}/a)^{2} + 2(s ‒ s_{0})/a] |
θ = θ_{0} + ω_{0}t + ½αt^{2} | φ = (-β/ψ_{0}) +
√[(β/ψ_{0})^{2} + 2β(s ‒ s_{0})] |
Third Equation of Space-Time | v² = v_{0}² + 2a(s – s_{0}) | s = s_{0} + (v² ‒ v_{0}²)/2a | ω² = ω_{0}² + 2α(θ – θ_{0}) | θ = θ_{0} + (ω^{2} ‒ ω_{0}^{2})/2α |
Distimement | s = (t ‒ t_{0})v | t = t_{0} + us | s = (φ ‒ φ_{0})ωR_{s}^{2} | φ = φ_{0} + ψs |
First Equation of Time-Space | 1/v = (1/v_{0}) + (s/a) | u = u_{0} + bs | s = (ψ ‒ ψ_{0})/β | ψ = ψ_{0} + βs |
Second Equation of Time-Space | s = (-u_{0}/b) +
√[( u_{0}/b)^{2} + 2(t ‒ t_{0})/b] |
t = t_{0} + u_{0}s + ½bs² | θ = (-α/ω_{0}) +
√[(α/ω_{0})^{2} + 2α(t ‒ t_{0})] |
φ = φ_{0} + ψ_{0}t + ½βs^{2} |
Third Equation of Time-Space | t = t_{0} + (u^{2} ‒ u_{0}^{2})/2b | u² = u_{0}² + 2b(t – t_{0}) | φ = φ_{0} + (ψ^{2} ‒ ψ_{0}^{2})/2β | ψ² = ψ_{0}² + 2β(φ – φ_{0}) |