Article — Terminal Velocity Calculator
Terminal Velocity Calculator: Drag, Gravity, and Limit Speed
Terminal velocity is the constant falling speed reached when drag force equals weight. The formula is vt = √(2mg / (ρCdA)). A belly-down skydiver hits about 55 m/s (200 km/h). A small raindrop reaches only 4 m/s. The cap on speed comes from air resistance growing as v².
The concept comes straight from Newton's second law. A falling object accelerates at g until air drag — proportional to v² — climbs to match its weight. Past that point, net force is zero and the object cruises at constant speed for the rest of the fall.
What is terminal velocity?
An object dropped in a vacuum accelerates indefinitely at 9.8 m/s². In air, drag fights gravity. Drag rises with speed squared, so the object accelerates fast at first, more slowly as it speeds up, and asymptotically approaches a limit at which drag exactly cancels weight. That limit is the terminal velocity.
Terminal velocity depends on four properties of the falling object and the fluid around it: mass, frontal area, shape (captured by the drag coefficient), and fluid density. Change any one and the limit shifts. Doubling mass raises terminal velocity by √2 ≈ 1.41. Doubling area cuts it by the same factor.
The terminal velocity formula in plain math
At terminal velocity, drag force ½ρv²CdA equals weight mg. Solving for v gives the standard equation.
v_t = √(2mg / (ρ·Cd·A)) main formulaF_drag = ½·ρ·v²·Cd·A drag forceτ = v_t / g characteristic time~2.5 τ ≈ time to reach 95% v_tFor a 80-kg skydiver with Cd = 1.0 and A = 0.5 m² in sea-level air (ρ = 1.225 kg/m³): vt = √(2 × 80 × 9.81 / (1.225 × 1.0 × 0.5)) = √(2562) ≈ 50.6 m/s. That is 182 km/h — close to the textbook value of 200 km/h for a belly-down skydiver.
Terminal velocity for skydivers
Body position controls a skydiver's terminal velocity. In the standard belly-down arched posture, the frontal area is about 0.5 m² and the drag coefficient is roughly 1.0, giving vt ≈ 55 m/s (200 km/h). Pulling arms and legs in tight reduces both A and Cd, and head-down freefall can exceed 90 m/s (320 km/h). Wingsuits and tracking suits do the opposite — they raise A enough to push vt down to 30 m/s while adding horizontal speed.
Terminal velocity of raindrops
Raindrops reach terminal velocity within the first few meters of fall because they are tiny and light. A 1-mm drop limits at about 4 m/s (14 km/h). A 5-mm drop reaches 9 m/s (32 km/h) — past that diameter, drops oscillate and break apart, so 5 mm is roughly the maximum drop size in nature.
If raindrops fell from a vacuum-perfect 1-km altitude, they would hit the ground at over 140 m/s (500 km/h) — fast enough to cause serious injury. Air resistance keeps actual impact speeds under 10 m/s, which is why standing in rain is unpleasant but harmless.
How altitude affects terminal velocity
Air density falls roughly exponentially with altitude. At 10 km (commercial jet cruise), density is about 0.41 kg/m³ — one-third of sea level. Since terminal velocity is inversely proportional to √ρ, a skydiver who reaches 55 m/s at sea level would reach 95 m/s at 10 km. At 39 km — Felix Baumgartner's exit altitude — density is one percent of sea level, and his terminal velocity exceeded the local speed of sound at Mach 1.25.
The atmospheric density drop is the reason high-altitude jumps require pressure suits and oxygen: not only is the air too thin to breathe, it is also too thin to brake against. Stratos team engineers had to design exit, freefall, and parachute deployment all around the changing vt.
Drag coefficient and the shape of falling objects
The drag coefficient Cd wraps up everything about how a shape interacts with the fluid into a single dimensionless number. Streamlined airfoils sit near 0.04. A sphere is 0.47. A flat plate facing the flow is 1.28. An open parachute is 1.4. Cd depends weakly on Reynolds number, so values can shift at very low or very high speeds.
- Streamlined teardrop = 0.04 — racing cars, dolphins
- Smooth sphere = 0.47 — billiard ball, marble
- Dimpled golf ball = 0.25 — turbulent boundary layer cuts wake
- Cone, point first = 0.5 — typical raindrop shape
- Cylinder, broadside = 1.15 — falling pencil sideways
- Flat plate, face-on = 1.28 — falling tray
- Parachute, fully open = 1.4 — designed for maximum drag
Terminal velocity records and milestones
The standing record for unpowered human terminal velocity belongs to Alan Eustace, the Google executive who in 2014 reached 1322 km/h (367 m/s, Mach 1.23) from 41,419 m. Felix Baumgartner held the record briefly in 2012 with 1357 km/h from 39 km. Both jumps depended on the same physics: thin atmospheric density at altitude raised vt high enough to break the local speed of sound.
The characteristic time τ = vt/g tells you how quickly an object approaches terminal velocity. For a skydiver, τ ≈ 5.6 s; the asymptote is essentially reached after 2 to 3 τ (about 15 seconds, or 350 to 500 m of fall). Raindrops have τ under 1 s and reach vt almost immediately.
Common terminal velocity mistakes
Using density in g/cm³ instead of kg/m³ (off by 1000), using mass in pounds without converting to kilograms, treating Cd as constant when it varies with Reynolds number, ignoring altitude when calculating skydiver vt, and confusing terminal velocity with the speed achieved in finite-distance free fall.
The most frequent error is forgetting that drag scales with v². A 50-percent increase in speed produces a 125-percent increase in drag. That is why parachutes work — once deployed, the huge area pushes vt down to 5 m/s within seconds. A second common error is using the wrong frontal area: the area is the cross-section perpendicular to the velocity, not the total surface area of the object.
The third pitfall is assuming any falling object reaches terminal velocity. Heavy, compact objects (cannonball, bowling ball, skydiver without parachute over short distances) can fall hundreds of meters before approaching vt. Calculate the time to vt with τ = vt/g before assuming the limit applies — for cannon-ball-style projectiles, finite-distance kinematics often gives a closer answer.