Mechanics & fluids

v = Δx/Δt · a = Δv/Δt

Both are vectors, so direction counts

v = v₀ + at · x = v₀t + ½at² · v² = v₀² + 2aΔx

Constant acceleration only

F_net = ma

Zero net force means constant velocity, not rest

f_s ≤ μ_s N · f_k = μ_k N

Static is a maximum; kinetic is a fixed value

F_parallel = mg sin θ · N = mg cos θ

Acceleration on a frictionless incline is g sin θ

a_c = v²/r · F_c = mv²/r

Points toward the center and does no work

τ = rF sin θ

Maximum when the force is perpendicular to the lever arm

W = Fd cos θ

Zero when force is perpendicular to displacement

KE = ½mv²

Doubling speed quadruples the energy

U_grav = mgh · U_spring = ½kx²

Only the height CHANGE matters

P = W/t = Fv

The watt is a joule per second

p = mv · J = FΔt = Δp

Momentum is conserved in every collision

F = −kx

The minus sign means the force opposes displacement

efficiency = W_out / E_in × 100%

Never 100% for a real heat engine

ρ = m/V · SG = ρ / ρ_water

Water is 1000 kg/m³, so SG > 1 sinks

P = F/A

The pascal is a newton per square meter

P = P₀ + ρgh

Depends on depth only, never on container shape

F_b = ρ_fluid V_displaced g

A floating object displaces its own weight

F₁/A₁ = F₂/A₂

The hydraulic lift; work is still conserved

A₁v₁ = A₂v₂

A narrower pipe means faster flow

P + ½ρv² + ρgh = constant

Faster flow means lower pressure

Q = πΔPr⁴ / 8ηL

Flow depends on the FOURTH power of the radius

P = 2γ/r

Small alveoli would collapse without surfactant