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 |
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Linear w/3D Space | Linear w/3D Time | Angular w/3D Space | Angular w/3D Time | |
Position | Linear length: s(t) | Linear time: t(s) | Angular length: θ(s) | Angular time: φ(t) |
Instantaneous Velocity/ Legerity | Linear velocity
v = ds/dt = R/Q = S/T v_{R} = dr/dt, v_{T} = ωR |
Linear legerity
u = dt/ds = Q/R = T/S u_{R} = dw/ds, u_{T} = ψQ |
Angular velocity
ω = dθ/dt = dt/dφ ω = v_{R}/R = 2π/T |
Angular legerity
ψ = dφ/ds = ds/dθ ψ = u_{R}/Q = 2π/S |
Instantaneous Acceleration/ Expedience |
Acceleration
a = dv/dt := 1/b |
Expedience
b = du/ds := 1/a |
Angular acceleration
α = dω/dt |
Angular expedience
β = dψ/ds |
Radial/Tangential Acceleration/ Expedience |
Acceleration
a_{R} = v^{2}/R = v/Q a_{T} = αR |
Expedience
b_{R} = u^{2}/Q = u/R b_{T} = βQ |
Angular acceleration
a_{R} = v^{2}/R = ω^{2}R α = a_{T}/R |
Angular expedience
b_{R} = ψ^{2}Q = u^{2}/Q β = b_{T}/Q |
Uniform Rates | v = 2πR/T | u = 2πQ/S | v_{T} = ωR | u_{T} = ψQ |
Radius | Spatial radius
R = S/(2π) = vQ |
Temporal radius
Q = T/(2π) = uR |
Spatial radius
R = s/θ = v/ω = 1/ψ |
Temporal radius
Q = t/φ = u/ψ = 1/ω |
Circumference
Path Length |
Circumference
S = 2πvQ = 2πR |
Circumference
S = 2πQ/u = 2πR |
Spatial path length
θ = s/R |
Temporal path length
S = 2π/ψ |
Period | T = 2πR/v = 2πQ | T = 2πuR = 2πQ | T = 2π/ω | φ = t/Q |
Travel Length/Time | s = vt = ωt R | t = us = ψs/Q | θ = s/(t R) = v/R | ψ = t/s Q = u Q |
Displacement | s = s_{0} + vt | t = (s ‒ s_{0})u | θ = θ_{0} + ωt | t = (θ ‒ θ_{0})/ω |
First Equation of Space-Time | v = v_{0} + at | 1/u = (1/u_{0}) + (t/b) | 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})/ψ | 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/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/ Invertegy | Kinetic Energy: E = ½mv^{2} | Kinetic invertegy: V = ½nu^{2} | Rotational KE: ½Iω^{2} | Rotational KE: ½Jψ^{2} |
Force/Rush | Force: F = ma | Rush: Γ = nb | Torque: τ = Iα | Strophence: σ = Jβ |
Work/Invork | Linear work: W = Fs | Linear invork: V = Γ t | Rotational work: W = τθ | Rotational invork: V = σφ |
Power/Exertion | Linear power: Fv | Linear exertion: Γu | Rotational power: τω | Rotational exertion: σψ |