Terminal Velocity

The setup

An object falls in air. Forces:

• Gravity — F_g = mg (constant, downward).
• Air drag — F_drag = ½·ρ·C_d·A·v² (upward, opposing motion).

where ρ is air density, C_d is drag coefficient (depends on shape), A is cross-sectional area perpendicular to motion, v is velocity.

At terminal velocity, F_g = F_drag — net force is zero, no further acceleration:

m·g = ½·ρ·C_d·A·v_t²
v_t = √(2mg / (ρ·C_d·A))

Common terminal velocities

Object	Approx. v_t	Notes
Mist droplet (~0.01 mm)	0.001 m/s	Almost suspended
Drizzle drop (0.5 mm)	~2 m/s	Light rain
Heavy raindrop (~3 mm)	~9 m/s	Steady rain
Hail (1 cm diameter)	~14 m/s	Damaging hailstones
Snowflake	~1 m/s	Very low because of large surface
Ping-pong ball (no spin)	~9 m/s	Light + small
Bowling ball	~80-90 m/s	Heavy + smooth
Skydiver (spread-eagle)	~50 m/s (115 mph)	Max area
Skydiver (head-first)	~90 m/s (200 mph)	Min area
Penny dropped from skyscraper	~25 m/s	Mythbusters: not lethal, just stings
Bullet (0.45 ACP)	~46 m/s	Falling straight down
Felix Baumgartner (39 km altitude)	377 m/s	Above sound barrier

Approach to terminal velocity

Object starts at rest, accelerates under gravity. Drag increases as v². Eventually:

Phase	v	Behavior
Initial fall	v ≪ v_t	Mostly gravity; a ≈ g
Mid-fall	v ~ ½·v_t	Drag becoming significant
Approach to terminal	v → v_t	Drag nearly balances gravity
Terminal	v = v_t	Constant speed

Mathematically (with quadratic drag):

v(t) = v_t · tanh(g·t / v_t)

So v reaches ~99% of v_t after t ≈ 2.65·v_t/g. For a skydiver (v_t = 53 m/s), this is ~14 seconds — and ~370 m of fall.

JavaScript — terminal velocity calculations

// Terminal velocity from physical parameters
function terminalVelocity(mass, area, dragCoeff = 1.0, airDensity = 1.225) {
  return Math.sqrt(2 * mass * 9.81 / (airDensity * dragCoeff * area));
}

// Skydiver
console.log(`Spread-eagle: ${terminalVelocity(75, 0.7, 1.0).toFixed(1)} m/s`); // ~50
console.log(`Head-first:   ${terminalVelocity(75, 0.18, 0.7).toFixed(1)} m/s`); // ~90

// Raindrop (1mm radius, water density 1000 kg/m³)
const r = 0.001;  // 1 mm
const m_drop = (4/3) * Math.PI * Math.pow(r, 3) * 1000;
const A_drop = Math.PI * r * r;
console.log(`1mm raindrop: ${terminalVelocity(m_drop, A_drop, 0.5).toFixed(1)} m/s`); // ~6

// Numerical simulation: position vs time with drag
function simulateFall(mass, area, dragCoeff = 1.0, airDensity = 1.225, dt = 0.01, max_time = 30) {
  let v = 0, y = 0;
  const data = [{ t: 0, v: 0, y: 0 }];
  for (let t = dt; t < max_time; t += dt) {
    const dragForce = 0.5 * airDensity * dragCoeff * area * v * v;
    const F = mass * 9.81 - dragForce;  // gravity minus drag
    v += (F / mass) * dt;
    y += v * dt;
    if (Math.abs(t * 100 - Math.round(t * 100)) < dt) {
      data.push({ t, v, y });
    }
  }
  return data;
}

// Skydiver fall
const skydive = simulateFall(75, 0.7, 1.0);
const v_t_actual = terminalVelocity(75, 0.7, 1.0);
const data99 = skydive.find(d => d.v > 0.99 * v_t_actual);
console.log(`Skydiver reaches 99% of terminal v at t = ${data99?.t.toFixed(1)} s`);

// Analytical: v(t) = v_t · tanh(g·t/v_t)
function vAtTime(t, v_t, g = 9.81) {
  return v_t * Math.tanh(g * t / v_t);
}

console.log(`Skydiver at t=5s: ${vAtTime(5, v_t_actual).toFixed(1)} m/s`); // close to terminal already
console.log(`Skydiver at t=15s: ${vAtTime(15, v_t_actual).toFixed(1)} m/s`); // ~99.99% of terminal

// At what altitude does air density change v_t much?
function densityAtAltitude(altitude_m) {
  // Simple exponential: ρ(h) = ρ_0 * exp(-h/H)
  const H = 8000;  // scale height
  return 1.225 * Math.exp(-altitude_m / H);
}

// At 39 km altitude, density is...
console.log(`Density at 39 km: ${densityAtAltitude(39000).toExponential(2)} kg/m³`); // very thin
const v_t_39km = terminalVelocity(75, 0.18, 0.7, densityAtAltitude(39000));
console.log(`Terminal v at 39 km: ${v_t_39km.toFixed(0)} m/s`);  // very high

Where terminal velocity matters

• Skydiving and parachuting. Terminal velocity in different postures; parachute design for safe landing speeds.
• Atmospheric science. Raindrop and hail formation, air pollution settling rates, snow/cloud particle dynamics.
• Ballistics. Falling bullets, dropped objects from buildings — how dangerous? (Spoiler: pennies from the Empire State don't kill you).
• Aeronautics. Aircraft drop tests, parachute system design, ejection seat physics.
• Reentry physics. Spacecraft heating during atmospheric reentry — at orbital speed, "terminal velocity" is irrelevant; spacecraft must slow via aerodynamic friction (heat shields).
• Sport science. Falling objects in BASE jumping, wingsuit flight (lift in addition to drag).
• Geology. Sediment deposition rates in fluids — Stokes' law version for low Reynolds (small particles).

Common mistakes

• Thinking heavier objects always fall faster. True only when drag is significant. In vacuum, all objects fall at g. With drag, terminal v depends on mass/area ratio.
• Forgetting drag is quadratic in v at high speeds. Linear drag (F = -bv) only applies at very low speeds (small particles in fluids, "Stokes drag"). High speeds: F ~ -v²·ρ·C_d·A.
• Assuming terminal velocity is reached instantly. Takes time — for a skydiver, ~12-15 seconds and ~300-400 m. Many fall scenarios end before reaching terminal.
• Ignoring shape effects. Drag coefficient C_d varies hugely — flat plate ~1.28, sphere ~0.47, smooth car ~0.3, optimized cyclist ~0.17. Shape matters.
• Forgetting air density change with altitude. At 10 km up, air is ~1/3 of sea-level density; terminal velocity is √3 times higher.
• Confusing terminal velocity with maximum velocity. "Maximum" usually means top speed reached. Terminal velocity is the asymptotic limit. They're equal only after enough fall distance.