Expanding on a previous post *here*, this is a summary of the equations of motion for space-time and time-space. See also a pdf version in the *Time-space Glossary* option above.

*s* = displacement magnitude, *t* = time magnitude, *v* = velocity, *v*_{0} = initial velocity, *a* = acceleration, *ℓ* = celerity, *ℓ _{0}* = initial celerity,

*b*= deprestination,

*ω*= angular velocity,

*ω*

_{0}= initial angular velocity,

*ψ*= angular celerity,

*ψ*

_{0}= initial angular celerity,

*θ*= spatial angle,

*ψ*= temporal angle, S = circumference, T = period, R

_{s}= spatial radius, and R

_{t}= temporal radius.

Linear w/3D space | Linear w/3D time | Angular w/3D space | Angular w/3D time | |

Average Rate | v = Δs/Δt |
ℓ = Δt/Δs |
ω = Δθ/Δt |
ψ = Δϑ/Δs |

Average 2^{nd} Rate |
a = Δv/Δt |
b = Δℓ/Δs |
α = Δω/Δt |
β = Δψ/Δs |

Instantaneous Rate | Velocityv = ds/dt |
Celerity ℓ = d t/ds |
Angular velocityω = dθ/dt = dt/dϑ |
Angular celerityψ = dϑ/ds = ds/dθ |

Instantaneous 2^{nd} Rate |
Accelerationa = dv/dt |
Deprestinationb = dℓ/ds |
Tangential accelerationα = dω/dt |
Tangential deprestinationβ = dψ/ds |

Centripetal/Radial 2^{nd} Rate |
Centripetal accelerationa = _{cen}v/R^{2}_{s} |
Centripetal deprestinationb = 1/(ℓ_{cen}^{2}R_{s}) |
Radial accelerationa = R_{rad}_{s} ω^{2} |
Radial deprestinationb = R_{rad}_{t} ψ^{2} |

Uniform Tangential Rate | v = 2πR_{tan}_{s}/T |
ℓ = T/(2πR_{tan}_{t}) |
v = R_{tan}_{s} ω |
ℓ = R_{tan}_{t} ψ |

Circumference/Arc Length | Spatial circumference S = 2πR _{s} |
Temporal circumference T = 2πR _{t} |
Spatial arc lengthθ = s/R_{s} |
Temporal arc lengthϑ = t/R_{t} |

Period | T = 2πR_{s}/v |
T = 2πR_{t}ℓ |
T = 2π/ω |
T = 2π/ψ |

Radius | Spatial radius R _{s} = S/(2πv) |
Temporal radius R _{t} = T/(2πℓ) |
Spatial radius R _{s} = ds/dθ = s/θ = v/ω |
Temporal radius R _{t} = dt/dϑ = t/ϑ = ℓ/ψ |

Position | s |
t |
On a circle: s = R_{s} θ |
On a cycle: t = R_{t} ϑ |

Displacement | s = s_{0} + vt |
t = tℓ_{0} + s |
θ = θ_{0} + ωt |
ϑ = ϑ_{0} + ψs |

Second Equation of Motion | s = s_{0} + v_{0}t + ½at² |
t = tℓ_{0} + _{0}s + ½bs² |
θ = θ_{0} + ω_{0}t + ½αt^{2} |
ϑ = ϑ_{0} + ψ_{0}t + ½βs^{2} |

First Equation of Motion | v = v_{0} + at |
ℓ = ℓ_{0} + bs |
ω = ω + αt |
ψ = ψ_{0} + βs |

Third Equation of Motion | v² = v(_{0}² + 2as – s)_{0} |
ℓ² = ℓ_{0}² + 2b(t – t)_{0} |
ω² = ω(_{0}² + 2αθ – θ)_{0} |
ψ² = ψ + 2_{0}²β(ϑ – ϑ)_{0} |