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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/Celerity Linear Velocity

v = ds/dt = 1/u

Celerity

u = dt/ds = 1/v

Angular velocity

ω = dθ/dt = dt/dφ

Angular celerity

ψ = dφ/ds = ds/dθ

Instantaneous Rate 2 Acceleration

a = dv/dt := 1/b

Prestination

b = du/ds := 1/a

Tangential acceleration

α = dω/dt

Tangential prestination

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/Alacrity Mass (linear inertia): m Vass (linear alacrity): n Rotational Inertia: I = mr2 Rotational Alacrity: J = nt2
Momentum/Celentum Momentum: p = mv Celentum: q = nu Angular Momentum: L = Iω Angular Celentum: Λ = Jψ
Kinetic Energy/Visity Kinetic Energy: E = ½mv2 Kinetic Visity: V = ½nu2 Rotational KE: ½2 Rotational KV: ½2
Force/Surge Force: F = ma Surge: Γ = nb Torque: τ = Iα Strophence: σ = Jβ
Work/Effort Linear Work: W = Fs Linear Effort: V = Γ t Rotational Work: W = τθ Rotational Effort: V = σφ
Power/Exertion Linear Power: Fv Linear Exertion: Γu Rotational Power: τω Rotational Exertion: σψ