How this instrument works
Drop anything into air and two forces argue. Weight pulls down at mg, constant. Aerodynamic drag pushes back at ½·C_d·ρ·A·v², starting from nothing and growing with speed squared. Somewhere those two match exactly, net force reaches zero, acceleration stops, so whatever is falling holds one steady speed for as long as air density and posture stay put. Terminal velocity is that speed, and rearranging that balance for v is all this instrument does.
Newton set out quadratic resistance in Book II of his Principia (1687), reasoning that any body sweeps out a tube of fluid and must shove aside mass proportional to ρ·A·v every second, each parcel carrying off momentum proportional to v. He tested it in 1710 by dropping glass spheres — some filled with air, some with mercury — from inside St Paul's Cathedral dome, timing falls of roughly 67 metres. What modern practice adds is C_d, an empirical bookkeeping factor absorbing everything geometry does to that idealised sweep: 0.47 for a smooth sphere, near 0.04 for a well-faired airship hull, 1.28 for a flat plate held broadside, roughly 1.0 for a skydiver spread belly-to-earth, 1.5 or more beneath a round canopy.
Three assumptions sit underneath. C_d is treated as fixed, when truthfully it tracks Reynolds number — a sphere's value collapses from about 0.5 toward 0.1 across a narrow band near Re ≈ 3×10⁵, once its boundary layer turns turbulent, which is precisely what golf ball dimples exploit. Buoyancy is neglected, harmless for steel in air where displaced mass is a rounding error, badly wrong for grit settling in water or for balloons. Air density, too, enters as one fixed number, whereas jumpers leaving 4 km meet ρ ≈ 0.82 kg/m³, fall faster than any sea-level figure predicts, then decelerate as thicker air lower down catches up.
- Mass of the falling object takes grams, kilograms or pounds — enter total falling mass, gear and clothing included, not just some payload figure.
- Drag coefficient is a bare number tied to shape and orientation: 0.47 sphere, 1.0 skydiver belly-to-earth, 1.28 flat plate broadside, 1.5–1.75 round parachute.
- Air density defaults to 1.225 kg/m³, standard sea level at 15 °C. Use 1.0 near 2 km, 0.82 near 4 km, or 998 kg/m³ if your object is sinking through water instead.
- Frontal area means projected area facing airflow — silhouette cast by a lamp directly ahead, never total surface area.
- Read Terminal velocity in m/s, km/h or mph, then treat it as a ceiling approached gradually rather than some speed reached at once.
Worked example — a skydiver at full spread
An 80 kg jumper, gear included, lying flat and stable. Mass of the falling object reads 80 kg; Drag coefficient 1.0, fair for so bluff a posture; Air density 1.225 kg/m³ for sea-level air; Frontal area 0.7 m² for a spread-eagled adult. Then v_t = √(2 × 80 × 9.80665 ⁄ (1.0 × 1.225 × 0.7)) = √(1569.064 ⁄ 0.8575) = √1829.81 = 42.78 m/s — 154 km/h, or 95.7 mph.
Parachuting manuals usually quote 120 mph for this position, and that gap instructs rather than embarrasses: 0.7 m² is a generous, fully relaxed spread. Tuck limbs inward toward 0.44 m² and this sheet returns 53.9 m/s, which is 120 mph almost exactly. Go head-down at 0.175 m² and it reads 85.6 m/s, near 190 mph — double our first answer, because area sits under a square root, so quartering one doubles another. Notice too how slowly any ceiling arrives: v_t ⁄ g works out at 4.4 seconds, and 99% of terminal needs roughly 2.6 such time constants, meaning some 11.5 seconds and 365 metres of sky.
Questions
Do heavier objects really fall faster?
Once drag matters, yes — but only through √m, and only at fixed shape. Double mass behind an identical outline and steady speed climbs just 41%. Geometric scaling bites harder: shrink something without altering its density and mass drops as length cubed while frontal area drops as length squared, so v_t scales with √length. J. B. S. Haldane put it memorably in 1926 — drop a mouse down a mine shaft and it walks away, drop a horse and it splashes.
Which area do I enter — frontal, surface, or plan?
Frontal: projected area normal to airflow. For spheres of radius r that means πr², not their 4πr² skin; for jumpers, silhouette viewed from below. Substituting total surface area is far and away the commonest error here, and it understates answers badly. Remember also that C_d and A travel as a matched pair — every tabulated coefficient is defined against one specific reference area, so pairing wetted-area values with frontal figures yields nonsense.
Is the drag coefficient genuinely constant?
No, it tracks Reynolds number, and this sheet holds it fixed. Across ordinary falling speeds that approximation serves blunt shapes well: a sphere hovers near 0.47 over several decades of Re. Near Re ≈ 3×10⁵, though, its boundary layer turns turbulent, separation shifts rearward, C_d collapsing toward 0.1 — briefly, a smooth sphere gets slipperier as it speeds up. Golf ball dimples trip that transition early and deliberately.
How much does altitude change my answer?
Through air density alone, inversely and under a square root: halve ρ and steady speed climbs 41%. Standard atmosphere supplies 1.225 kg/m³ at sea level, 0.82 at 4 km, 0.41 at 10 km. That dependence explains stratospheric jumps — Felix Baumgartner passed 377 m/s at 39 km in October 2012, faster than sound, through air near one per cent of sea-level density, then slowed steadily as thicker layers arrived.
Should buoyancy be subtracted from the weight?
Rarely in air, where displaced mass runs thousandths of a dense object's own. Elsewhere it counts: swap mg for (m − ρV)g, taking off displaced fluid. Sand settling in water, rising bubbles, helium balloons — all need that correction, and anything less dense than its surroundings ends up with steady speed pointing upward instead of down.
When does this quadratic drag law stop applying?
At small size and low speed, where flow stays laminar and resistance grows in proportion to v rather than v². Below Reynolds numbers near 1 — fog droplets, pollen, settling silt — Stokes' regime takes over and steady speed follows v = 2r²g(ρ_p − ρ_f) ⁄ 9η, linear in weight, governed by viscosity rather than density. Between those regimes, roughly 1 to 1000 in Re, neither expression is exact; empirical drag charts remain the honest tool.