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More equations of motion

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, v0 = initial velocity, a = acceleration, u = celerity, u0 = initial celerity, b = modulation, ω = angular velocity, ω0 = initial angular velocity, ψ = angular celerity, ψ0 = initial angular celerity, θ = spatial angle, ψ = temporal angle, S = circumference, T = period, Rs = spatial radius, and Rt = temporal radius.

Linear w/3D space Linear w/3D time Angular w/3D space Angular w/3D time
Average Rate v =  Δst u = Δts ω = Δθt ψ = Δϑs
Average 2nd Rate a = Δvt b = Δu/Δs α = Δωt β = Δψs
Instantaneous Rate Velocity
v = ds/dt
Celerity
u = dt/ds
Angular velocity
ω = dθ/dt = dt/dϑ
Angular celerity
ψ = dϑ/ds = ds/dθ
Instantaneous 2nd Rate Acceleration
a = dv/dt
Modulation
b = du/ds
Tangential acceleration
α = dω/dt
Tangential modulation
β = dψ/ds
Centripetal/Radial 2nd Rate Centripetal acceleration
acen = v2/Rs
Centripetal modulation
bcen = 1/(u2Rs)
Radial acceleration
arad = Rs ω2
Radial modulation
brad = Rt ψ2
Uniform Tangential Rate vtan = 2πRs/T utan = T/(2πRt) vtan = Rs ω tan = Rt ψ
Circumference/Arc Length Spatial circumference
S = 2πRs
Temporal circumference
T = 2πRt
Spatial arc length
θ = s/Rs
Temporal arc length
ϑ = t/Rt
Period T = 2πRs/v T = 2πRtu T = 2π/ω T = 2π/ψ
Radius Spatial radius
Rs = S/(2πv)
Temporal radius
Rt = T/(2πu)
Spatial radius
Rs = ds/dθ = s/θ = v/ω
Temporal radius
Rt = dt/dϑ = t/ϑ = ℓ/ψ
Position s t On a circle: s = Rs θ On a cycle: t = Rt ϑ
Displacement s = s0 + vt t = t0 + us θ = θ0 + ωt ϑ = ϑ0 + ψs
Second Equation of Motion s = s0 + v0t + ½at² t = t0 + u0s + ½bs² θ = θ0 + ω0t + ½αt2 ϑ = ϑ0 + ψ0t + ½βs2
First Equation of Motion v = v0 + at u = u0 + bs ω = ω­ + αt ψ = ψ0 + βs
Third Equation of Motion = v0² + 2a(s – s0) u² = u0² + 2b(t – t0) ω² = ω0² + 2α(θ – θ0) ψ² = ψ0² + 2β(ϑ – ϑ0)

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