Kinematics (Constant Acceleration)
| Equation |
Variables |
Solves for |
| v = v₀ + a·t |
v₀=initial vel, a=accel, t=time |
Velocity at time t |
| x = x₀ + v₀·t + ½a·t² |
x₀=initial pos |
Position at time t |
| v² = v₀² + 2a·(x-x₀) |
|
Velocity without time |
| x = x₀ + ½(v₀+v)·t |
|
Displacement (avg velocity) |
Newton's Laws & Force
| Name |
Equation |
Notes |
| Newton's 2nd Law |
F = m·a |
a = F/m; sum all forces |
| Gravity (universal) |
F = G·m₁·m₂ / r² |
G = 6.674×10⁻¹¹ N·m²/kg²; attractive, along r̂ |
| Surface gravity |
g = G·M / r² |
Earth: g ≈ 9.81 m/s² |
| Impulse-Momentum |
J = Δp = m·Δv = F·Δt |
Instantaneous velocity change: Δv = J/m |
| Elastic collision |
v₁' = ((m₁-m₂)v₁ + 2m₂v₂) / (m₁+m₂)
|
Energy and momentum conserved |
| Friction |
F_f = μ·N |
μₛ (static) > μₖ (kinetic); N = normal force |
Rotational Dynamics
| Quantity |
Equation |
Analogue |
| Torque |
τ = r × F = I·α |
F = m·a |
| Moment of inertia (solid sphere) |
I = 2/5 · m·r² |
mass |
| Moment of inertia (thin rod, centre) |
I = 1/12 · m·L² |
|
| Angular momentum |
L = I·ω = r × p |
p = m·v |
| Angular velocity → linear |
v = ω × r |
Cross product; tangential speed = ω·|r| |
| Centripetal acceleration |
a_c = ω²·r = v²/r |
Directed inward, toward centre |
| Gyroscopic precession |
Ω = τ / L |
Precession rate of a spinning gyroscope under torque τ
|
Spring & Damper
| Component |
Equation |
Notes |
| Hooke's Law (spring) |
F_s = -k·x |
k = spring constant; x = displacement from rest |
| Viscous damping |
F_d = -b·v |
b = damping coefficient; opposes velocity |
| Spring-damper ODE |
m·ẍ + b·ẋ + k·x = F(t) |
Standard 2nd-order linear ODE |
| Natural frequency |
ω₀ = √(k/m) |
Undamped oscillation frequency [rad/s] |
| Critical damping |
b_c = 2·√(k·m) |
ζ = b/b_c; ζ<1: underdamped, ζ=1: critical, ζ>1:
overdamped
|
| Damped frequency |
ω_d = ω₀·√(1-ζ²) |
Actual oscillation frequency when underdamped |
| 2D/3D spring (vector) |
F = -k·(|r|-L₀)·r̂ |
L₀ = rest length; r̂ = unit vector from anchor to end |
Orbital Mechanics
| Property |
Equation |
Notes |
| Circular orbit velocity |
v = √(G·M / r) |
Speed for stable circular orbit at radius r |
| Escape velocity |
v_esc = √(2·G·M / r) |
√2 × circular orbit speed |
| Orbital period (Kepler 3rd) |
T² = (4π²/G·M) · a³ |
a = semi-major axis |
| Vis-viva equation |
v² = G·M·(2/r - 1/a) |
Speed anywhere on elliptical orbit |
| Eccentricity |
e = √(1 + 2Eh²/(G²M²m³)) |
e=0 circle, 0<e<1 ellipse, e=1 parabola, e>1
hyperbola
|
Fluid & Drag
| Law / Force |
Equation |
Notes |
| Stokes drag (low Re) |
F_d = 6π·μ·r·v |
Sphere of radius r in fluid viscosity μ; laminar flow |
| Quadratic drag (high Re) |
F_d = ½·ρ·C_d·A·v² |
ρ=fluid density, C_d=drag coefficient, A=cross-section area
|
| Reynolds number |
Re = ρ·v·L / μ |
Re<2300 laminar, Re>4000 turbulent |
| Buoyancy (Archimedes) |
F_b = ρ_fluid · V_submerged · g
|
Upward; net force = F_b - m·g |
| Continuity (incompressible) |
A₁·v₁ = A₂·v₂ |
Conservation of mass in a pipe |
| Bernoulli |
p + ½·ρ·v² + ρ·g·h = const |
Energy conservation along streamline; explains lift |
| Navier-Stokes (incompressible) |
ρ(∂v/∂t + v·∇v) = -∇p + μ∇²v + f
|
Full fluid dynamics; f = external body forces |
In particle simulations, use Stokes drag for slow particles in
viscous media. Use quadratic drag for fast-moving objects
(projectiles, spacecraft). Quadratic drag is proportional to v², so
it becomes dominant at high speeds.