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/R_{s} | Angular duration: φ = t/R_{t} |
Average Rate | v = Δs/Δt | u = Δt/Δs | ω = Δθ/Δt = v/R_{s} | ψ = Δφ/Δs = u/R_{t} |
Average Rate 2 | a = Δv/Δt | b = Δu/Δs | α = Δω/Δt | β = Δψ/Δs |
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 = s_{0} + vt | t = (s ‒ s_{0})u | θ = θ_{0} + ωt | t = (θ ‒ θ_{0})ψR_{t}^{2} |
First Equation of Space-Time | v = v_{0} + at | t = (v ‒ v_{0})/a | ω = ω_{0} + αt | t = (ω ‒ ω_{0})/α |
Second Equation of Space-Time | s = s_{0} + v_{0}t + ½at² | t = (-u_{0}/a) +
√[(u_{0}/a)^{2} + 2(s ‒ s_{0})/a] |
θ = θ_{0} + ω_{0}t + ½αt^{2} | φ = (-β/ψ_{0}) +
√[(β/ψ_{0})^{2} + 2β(s ‒ s_{0})] |
Third Equation of Space-Time | v² = v_{0}² + 2a(s – s_{0}) | s = s_{0} + (v² ‒ v_{0}²)/2a | ω² = ω_{0}² + 2α(θ – θ_{0}) | θ = θ_{0} + (ω^{2} ‒ ω_{0}^{2})/2α |
Distimement | s = (t ‒ t_{0})v | t = t_{0} + us | s = (φ ‒ φ_{0})ωR_{s}^{2} | φ = φ_{0} + ψs |
First Equation of Time-Space | 1/v = (1/v_{0}) + (s/a) | u = u_{0} + bs | s = (ψ ‒ ψ_{0})/β | ψ = ψ_{0} + βs |
Second Equation of Time-Space | s = (-u_{0}/b) +
√[(u_{0}/b)^{2} + 2(t ‒ t_{0})/b] |
t = t_{0} + u_{0}s + ½bs² | θ = (-α/ω_{0}) +
√[(α/ω_{0})^{2} + 2α(t ‒ t_{0})] |
φ = φ_{0} + ψ_{0}t + ½βs^{2} |
Third Equation of Time-Space | t = t_{0} + (u^{2} ‒ u_{0}^{2})/2b | u² = u_{0}² + 2b(t – t_{0}) | φ = φ_{0} + (ψ^{2} ‒ ψ_{0}^{2})/2β | ψ² = ψ_{0}² + 2β(φ – φ_{0}) |
Inertia/Alacrity | Mass (linear inertia): m | Vass (linear alacrity): n | Rotational Inertia: I = mr^{2} | Rotational Alacrity: J = nt^{2} |
Momentum/Celentum | Momentum: p = mv | Celentum: q = nu | Angular Momentum: L = Iω | Angular Celentum: Λ = Jψ |
Kinetic Energy/Visity | Kinetic Energy: E = ½mv^{2} | Kinetic Visity: V = ½nu^{2} | Rotational KE: ½Iω^{2} | Rotational KV: ½Jψ^{2} |
Force/Trabence | Force: F = ma | Trabence: 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 |