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Motion Equations

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

  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/Rs Angular duration: φ = t/Rt
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
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 = 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)
Inertia/Facilia Mass (linear inertia): m Vass (linear facilia):  Rotational inertia: I = mr2 Rotational facilia: J = nt2
Momentum/Celentum Momentum: p = mv Celentum: q = nu Angular momentum: L = Iω Angular celentum: Λ = Jψ
Kinetic Energy/Epimony Kinetic Energy: KE = ½mv2 Kinetic Epimony: KE = ½nu2 Rotational KE: ½2 Rotational KE: ½2
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