Why does a soap bubble have higher excess pressure than a liquid drop of the same radius?

Soap film is flexible and has two surfaces (outer and inner) → Each surface has surface tension S acting inward → Force on a hemispherical surface = S × 2πR (tension along circumference) → Each surface contributes pressure = Force / Area = S × 2πR / (π$R^{2}$) = 2S/R → Two surfaces contribute twice → $\Delta P$_bubble = 4S/R → A liquid drop has only one surface (outer only) → $\Delta P$_drop = 2S/R → Therefore at the same R, bubble pressure > drop pressure by exactly a factor of 2.

Why does terminal velocity depend on $r^{2}$?

A sphere of radius r has weight ∝ $r^{3}$ (W = ^{4}⁄_{3}π$r^{3}$ρg) → Buoyancy ∝ $r^{3}$ as well (similar reasoning) → Net downward force ∝ $r^{3}$ → Stokes drag F = 6πηrv ∝ r × v → At terminal velocity: net force = drag → $r^{3}$ ∝ r × v_t → v_t ∝ $r^{3}$/r = $r^{2}$ → Hence doubling r quadruples v_t. This explains why large raindrops fall faster and why fine particles (r small) settle extremely slowly.

Why does Bernoulli's equation imply airplane lift?

Wing is cambered (curved on top, flatter below) → Air travels farther over top than below in same time → Top velocity v_top > Bottom velocity v_bottom (by continuity: constant flow rate) → Bernoulli: higher v → lower P → P_top < P_bottom → Net upward pressure difference × wing area = Lift force → Airplane stays airborne. The key step is: more distance on top → higher velocity → lower pressure → net lift.