The kinematic equations of motion have different forms depending on whether the motion is linear or angular (rotational) and whether space or time are 3D. They are given below and in a pdf here.
Parallel Equations of Motion |
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Linear w/3D space | Linear w/3D time | Angular w/3D space | Angular w/3D time | |
Position | Linear distance: s | Linear duration: t | Angular distance: θ = s/R_{s} | Angular duration: φ = t/R_{t} |
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 |
Velocity/Allegrity | Velocity
v = ds/dt = 1/u |
Allegrity
u = dt/ds = 1/v |
Angular velocity
ω = dθ/dt = dt/dφ |
Angular allegrity
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 |
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}) |
Inertia/Facilia | Mass (linear inertia): m | Vass (linear facilia): ℓ | Rotational inertia: I = mr^{2} | Rotational facilia: J = nt^{2} |
Momentum/Celentum | Momentum: p = mv | Celentum: q = nu | Angular momentum: L = Iω | Angular celentum: Λ = Jψ |
Kinetic Energy/Epimony | Kinetic Energy: KE = ½mv^{2} | Kinetic Epimony: KE = ½nu^{2} | Rotational KE: ½Iω^{2} | Rotational KE: ½Jψ^{2} |
Force/Surge | Force: F = ma | Surge: Γ = nb | Torque: τ = Iα | Strophence: σ = Jβ |
Work/Persistence | Linear work: W = Fs | Linear persistence: V = Γ t | Rotational work: W = τθ | Rotational persistence: V = σφ |
Power/Exertion | Linear power: Fv | Linear exertion: Γu | Rotational power: τω | Rotational exertion: σψ |