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, w = lenticity, w0 = initial lenticity, b = relentation, ω = angular velocity, ω0 = initial angular velocity, ψ = angular lenticity, ψ0 = initial angular lenticity, 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 = Δs/Δt | u = Δt/Δs | ω = Δθ/Δt | ψ = Δϑ/Δs |
| Average 2nd Rate | a = Δv/Δt | b = Δu/Δs | α = Δω/Δt | β = Δψ/Δs |
| Instantaneous Rate | Velocity v = ds/dt | Lenticity w = dt/ds | Angular velocity ω = dθ/dt = dt/dϑ | Angular lenticity ψ = dϑ/ds = ds/dθ |
| Instantaneous 2nd Rate | Acceleration a = dv/dt | Relentation b = dw/ds | Tangential acceleration α = dω/dt | Tangential relentation β = dψ/ds |
| Centripetal/Radial 2nd Rate | Centripetal acceleration acen = v2/Rs | Cenfugal relentation bcen = 1/(w2Rs) | Radial acceleration arad = Rs ω2 | Radial relentation brad = Rt ψ2 |
| Uniform Tangential Rate | vtan = 2πRs/T | utan = T/(2πRt) | vtan = Rs ω | ℓtan = 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 = 2πRtw | T = 2π/ω | T = 2π/ψ |
| Radius | Spatial radius Rs = S/(2πv) | Temporal radius Rt = T/(2πw) | 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 + ws | θ = θ0 + ωt | ϑ = ϑ0 + ψs |
| Second Equation of Motion | s = s0 + v0t + ½at² | t = t0 + w0s + ½bs² | θ = θ0 + ω0t + ½αt2 | ϑ = ϑ0 + ψ0t + ½βs2 |
| First Equation of Motion | v = v0 + at | w = w0 + bs | ω = ω + αt | ψ = ψ0 + βs |
| Third Equation of Motion | v² = v0² + 2a(s – s0) | w² = w0² + 2b(t – t0) | ω² = ω0² + 2α(θ – θ0) | ψ² = ψ0² + 2β(ϑ – ϑ0) |