Classical

Independence of motion (projectile)

Horizontal and vertical components of projectile motion are independent — solve as two separate 1D problems

Conservative force

A force where work done is path-independent, allowing a potential energy V(x) to be defined

Horizontal position (projectile, no horizontal force)

x(t) = v0·t

Vertical position (projectile, y-axis up)

y(t) = h − ½gt², with vy(t) = −gt

Time of flight formula

t_R = sqrt(2h/g)

Impact speed formula

v_R = sqrt(vx² + vy²)

Work done by gravity (falling height h)

W = mgh

Total mechanical energy (conservative field)

E = ½mv² + mgy = constant

Eigenfrequency ω0

The natural angular frequency of oscillation without friction; ω0 = sqrt(k/m)

Damping parameter γ

Constant characterizing the rate of amplitude decay due to friction; γ = b/m

Underdamped condition

ω0 > γ/2

Damped angular frequency

ω = sqrt(ω0² − (γ/2)²)

Critically damped condition

ω0 = γ/2

Heavily (over) damped condition

ω0 < γ/2, with α = sqrt((γ/2)² − ω0²)

Quality factor Q

Represents the percentage of energy lost per oscillation cycle; higher Q = better, less lossy oscillator. Q = ω0/γ

Energy decay of a DHO

E(t) = E0·e^(−γt) — note energy decays TWICE as fast as the amplitude

Angular momentum (central field)

L = mr²θ̇ = constant (conserved quantity in any central field)

Eccentricity ε meaning

Dimensionless orbit-shape parameter: ε=0 circle, 0

Perihelion / Aphelion

Perihelion (rp) = closest distance to the central body; Aphelion (ra) = furthest distance, in an orbit

Force constant k

k = GMm for a gravitational field, or k = e²/(4πε0) for an electrostatic (Coulomb) system

Total energy in a central field

E = ½m·ṙ² + L²/(2mr²) − k/r

Semi-major axis formula

a = (rmin + rmax)/2 = k/(2|E|)

Total orbital energy in terms of semi-major axis

E = −k/(2a)

Vis-viva equation

v² = GM(2/r − 1/a)

Escape velocity

v_esc = sqrt(2GM/r) (obtained by setting a → ∞ in the vis-viva equation)

Bohr model angular momentum quantization

L_n = n·ħ, for n = 1, 2, 3, …