Equations for circular/harmonic and spiral/helical motion in space and time is giver in a pdf here. The kinematic equations below have different forms depending on whether the motion is linear or angular (rotational) and whether space or time are 3D. They are also given in a pdf here.
Parallel Equations of Motion | ||||
Linear w/3D Space | Linear w/3D Time | Angular w/3D Space | Angular w/3D Time | |
Length | Time | Linear length: s | Linear time: t | Angular length: θ = s/r | Angular time: φ = t/q |
Velocity | Legerity | Linear/Tangential velocity
v_{T} = ds/dt = rω = r/q = S/T |
Linear/Tangential legerity
u_{T} = dt/ds = qψ = q/r = T/S |
Angular velocity
ω = dθ/dt = dt/dφ = v/r = 1/q |
Angular legerity
ψ = dφ/ds = ds/dθ = u/q = 1/r |
Acceleration | Expedience | Radial acceleration
a_{R} = v^{2}/r = rω^{2} = v/q = r/q² |
Radial expedience
b_{R} = u^{2}/q = qψ^{2} = u/r = q/r² |
Angular acceleration
α = dω/dt = a_{T}/r |
Angular expedience
β = dψ/ds = b_{T}/q |
Tangential acceleration
a_{T} = dv_{T}/dt = rα |
Tangential expedience
b_{T} = du_{T}/ds = qβ |
|||
Radius | Spatial radius
r = S/(2π) = qv |
Temporal radius
q = T/(2π) = ru |
Spatial radius
r = s/θ = v/ω = 1/ψ |
Temporal radius
q = t/φ = u/ψ = 1/ω |
Circumference | Period | S = 2πvq = 2πr | T = 2πur = 2πq | S = 2π/ψ | T = 2π/ω |
Revolutions | Repetitions
Frequency | Circuncy |
N = θ/(2π) | Z = φ/(2π) | f = ω/(2π) = 1/T | h = ψ/(2π) = 1/S |
Displacement | Distimement | s = s_{0} + vt | t = t_{0} + ux | θ = θ_{0} + ωt | φ = φ_{0} + ψs |
First Equation of Motion | v = v_{0} + a_{T}t | u = u_{0} + b_{T}s | ω = ω_{0} + αt | ψ = ψ_{0} + βs |
Second Equation of Motion | s = s_{0} + v_{0}t + ½a_{T}t² | t = t_{0} + u_{0}s + ½b_{T}s² | θ = θ_{0} + ω_{0}t + ½αt^{2} | φ = φ_{0} + ψ_{0}t + ½βs^{2} |
Third Equation of Motion | v² = v_{0}² + 2a(s – s_{0}) | u² = u_{0}² + 2b(t – t_{0}) | ω² = ω_{0}² + 2α(θ – θ_{0}) | ψ² = ψ_{0}² + 2β(φ – φ_{0}) |
Inertia | Facilia | Mass (linear inertia): m | Vass (linear facilia): n | Rotational inertia: I = mr^{2} | Rotational facilia: J = nt^{2} |
Momentum | Fulmentum | Momentum: p = mv | Fulmentum: q = nu | Angular momentum: L = Iω | Angular fulmentum: Λ = Jψ |
Kinetic Energy of Space or Time |
Kinetic space energy: E = ½mv^{2} | Kinetic time energy: V = ½nu^{2} | Rotational KSE: ½Iω^{2} | Rotational KTE: ½Jψ^{2} |
Force | Rush | Force: F = ma | Rush: Γ = nb | Torque: τ = Iα | Strophence: σ = Jβ |
Work | Effort | Linear work: W = Fs | Linear effort: V = Γt | Rotational work: W = τθ | Rotational effort: V = σφ |
Power | Alacrity | Linear power: P = Fv | Linear alacrity: Q = Γu | Rotational power: P = τω | Rotational alacrity: Q = σψ |