Terminal Velocity Calculator

Terminal velocity is the fastest speed a falling object can reach: the point where air resistance pushing up exactly balances weight pulling down, so the object stops accelerating and falls at a steady speed. Enter the mass, the area presented to the airflow, and the drag coefficient, and this calculator returns the terminal velocity in m/s, km/h, and mph, the time to reach 95% of it, the speed and distance after a fall time you choose, plus drag force and kinetic energy at terminal velocity.

One naming note: in hospitals, “fall rate” means something completely different, the number of patient falls per 1,000 patient days, a ward safety measure. That healthcare tool lives on the fall rate calculator page. This page is only about falling objects and the speed limit the air sets for them.

Quick Answer

How to use the terminal velocity calculator

Eight entries describe the fall: mass, frontal area, drag coefficient, the air, gravity, and two optional timing inputs. The defaults are a falling person; every other case on this page is a small edit away.

1. Enter the mass and pick its unit (kg, g, lb, or oz). Mass sets the weight that drag has to match.
2. Enter the cross-sectional area and pick its unit (m², cm², mm², ft², or in²). This is the silhouette shown to the airflow, not total surface area: about 0.7 m² spread flat, nearer 0.18 m² head-down.
3. Set the drag coefficient. Pick a shape preset to fill in a standard Cd, or choose Custom and type your own. The parachute preset uses 1.4, the middle of its usual 1.3–1.5 range; the human preset uses 0.8, inside the usual 0.6–1.0 spread.
4. Set the air density, or let altitude set it. Leave 1.225 kg/m³ for sea level, or type an altitude (m, km, ft, or mi) and the tool fills the density from its atmosphere model. Gravity defaults to 9.81 m/s².
5. Optionally set an initial velocity and a fall time. With a fall time set, the results add the speed reached and distance fallen by that moment; leave it at 0 and those two results show N/A.
6. Click Calculate. Read the headline speed, the verdict band (low below 15 m/s, high from 15 m/s, extreme from 60 m/s), and the detail rows. Copy or share from the results box; Reset restores the defaults.

The drag equation and the terminal velocity formula

Terminal velocity comes from setting two forces equal: weight pulling down and aerodynamic drag pushing up.

Here m is mass, g is gravitational acceleration, ρ (rho) is air density, A is cross-sectional area, Cd is the drag coefficient, and v0 is the starting speed. The algebra is the standard textbook route: NASA Glenn’s flight equations with drag set weight equal to drag for a coasting descent and solve for the same square-root expression, and OpenStax University Physics, in its drag force and terminal speed section, derives it the same way. Two features explain most real falls. Drag grows with the square of speed, so a falling object cannot accelerate forever. And because drag must equal a fixed weight at the balance point, the drag force this tool reports at terminal velocity always comes out as exactly m × g, a built-in check on the arithmetic.

The speed-after-time formula is the exact solution of the falling equation of motion under quadratic drag, written with hyperbolic functions, not a step-by-step simulation. Given long enough, tanh creeps towards 1 and the speed towards vt without ever mathematically arriving, which is why this tool reports the time to 95% rather than a time to “reach” terminal velocity. How speed builds under constant acceleration with no drag at all is what the acceleration calculator covers; drag bends the real fall away from that straight line.

Worked example: an 80 kg skydiver, belly to earth

The pre-filled case is a skydiver belly to earth: mass 80 kg, frontal area 0.7 m², drag coefficient 0.8 (the human preset), sea-level air at 1.225 kg/m³, gravity 9.81 m/s², dropped from rest, timed over 5 seconds.

1. Weight term: 2 × 80 × 9.81 = 1,569.6.
2. Drag term: 1.225 × 0.7 × 0.8 = 0.686.
3. Divide and take the root: 1,569.6 ÷ 0.686 = 2,288.05, and √2,288.05 = 47.83 m/s. In road units that is 172.20 km/h or 107.00 mph.
4. Time to 95%: (47.83 ÷ 9.81) × atanh(0.95) = 4.876 × 1.8318 = 8.93 seconds.
5. After 5 seconds: v = 47.83 × tanh(9.81 × 5 ÷ 47.83) = 36.93 m/s, with 105.72 m fallen. A no-drag fall would have covered 122.6 m in that time, so drag has already cost about 17 m of drop.
6. Forces and energy at the balance point: drag force = 784.80 N, which is exactly 80 × 9.81, the skydiver’s weight, and kinetic energy = ½ × 80 × 47.83² = 91,521.87 J.

Published skydiving figures put a typical belly-to-earth terminal velocity near 53 m/s, a little above this example. There is no contradiction: that figure assumes a slightly smaller area or a heavier jumper. The head-down card is the same person with only the posture changed, which is why speed skydivers tuck so tightly: area, not effort, is the lever.

Drag coefficient presets: what each shape is worth

The drag coefficient packs everything about a shape’s slipperiness into one dimensionless number. The presets are the standard reference values used across aerodynamics texts.

Cd never acts alone. It multiplies the area, so a parachute’s modest Cd advantage over a flat plate (1.4 against 1.28) is not what saves the skydiver; the roughly forty-fold jump in area is. Shape and size are separate dials: turn one at a time and watch which one your case depends on.

Why mass, area, and Cd pull in opposite directions

Mass raises terminal velocity, while area and drag coefficient lower it, and the square root softens every effect.

Start with mass. Drag at a given speed does not care how heavy the object is, but weight does: a heavier object needs more drag to balance it, and the only way to get more drag is to fall faster, until the v² term catches up. Because of the square root, the relationship is gentle: four times the mass gives twice the terminal velocity, and a 10% heavier skydiver with the same posture falls only about 5% faster. Area and Cd work the other way with the same softened strength: double the area and the speed falls to about 71% of what it was. The combination that moves fast is heavy with a small, slippery silhouette: a head-down skydiver, a hailstone. The combination that moves slowly is light, broad, and blunt: a leaf, a canopy. Terminal velocity is really a ranking of mass per unit of drag area.

Altitude and air density: thinner air, faster fall

Terminal velocity rises with altitude because air density falls. The altitude field models that fall with an exponential atmosphere: density = 1.225 × e^{−(M g h) ÷ (R T)}, using the molar mass of air, the gas constant, and a sea-level temperature of 288.15 K, the ideal-gas structure behind the Engineering ToolBox table of air temperature, pressure, and density against altitude.

This is why high-altitude jumpers fall fast at the top of a jump and slow as they descend into thicker air at a constant posture: terminal velocity is a moving target set by local density, a point NASA Glenn’s terminal velocity page makes when it notes that density varies with altitude and between planets. Air density itself is mass per unit volume, the same quantity the density calculator works out for solids and liquids. One caution: this tool holds the density you enter constant for the whole fall, so treat an altitude run as “terminal velocity in air of that density,” not a simulation of a fall through deepening air.

Time and distance: the approach is fast, then patient

Most of terminal velocity arrives quickly. The worked-example skydiver passes 36.93 m/s, about 77% of the final speed, in 5 seconds, reaches 95% at 8.93 seconds, and then spends the rest of the fall closing a shrinking gap.

The shape of that approach is set by one ratio, vt ÷ g, the fall’s time constant: about 4.88 seconds for the worked example. Light, draggy objects settle almost at once: the parachute case reaches 95% of its 5.52 m/s in 1.03 seconds. Sleek, heavy objects take longer: the head-down case needs 18.83 seconds to reach 95% of 100.84 m/s. The distance side of the same solution links to the acceleration and distance calculator, which covers distance covered while speed is still building. An initial velocity shifts the whole curve: starting at 30 m/s, the worked-example skydiver is at 45.09 m/s after 5 seconds and has covered 197.82 m, because the fall began partway up the curve.

Common misconceptions about terminal velocity

Three wrong ideas about falling speed survive every physics classroom. Each is a half-truth about the formula above.

• “Falling objects keep accelerating until they land.” Only in a vacuum. In air, acceleration starts at g and shrinks as drag builds, reaching zero at terminal velocity. A skydiver on a long jump spends most of it not accelerating at all.
• “Heavier objects always fall much faster.” Same shape and size, yes, but only with the square root of mass. Change the area too and the ranking flips: an 80 kg person under a canopy falls at 5.52 m/s while a 0.5 kg compact object (0.01 m², Cd 0.47) in this tool falls at over 40 m/s.
• “Terminal velocity is a fixed number for a person.” It is a number for a posture, a mass, and a density of air. One skydiver owns at least three on a single jump: about 100.84 m/s head-down, 47.83 m/s flat, and 5.52 m/s under canopy, before altitude moves all three.

What competing terminal velocity calculators miss

The rival tools for this phrase, Omni Calculator, OnlyCalculators, CalculatorDock, and Firgelli among them, all run the same square-root formula and the same skydiver example. The formula was never the gap. The gaps are around it:

• The time course. Most rival tools stop at the terminal velocity figure itself: no time to 95% of it, no speed and distance after a fall time you choose, which is what decides whether an object ever gets near its terminal velocity in the height available.
• Altitude as an input. Rivals generally ask for air density as a bare number and leave you to find it. This tool takes altitude directly and fills the density from an atmosphere model, in metres, kilometres, feet, or miles.
• The force and energy at the balance point. Drag force and kinetic energy at terminal velocity are standard follow-on questions, one extra multiplication each, yet routinely left to the reader. Here they are in the same result set, with the drag force doubling as a check: it must equal the weight.
• Honest preset ranges. Shape values are usually printed as single magic numbers. The presets here name the published ranges they sit inside (human 0.6–1.0, parachute 1.3–1.5), so a reader who needs the ends of the range can type them as a custom Cd.

Limitations of this model

This is the standard quadratic-drag model, and its limits are the model’s limits. It holds drag coefficient and air density constant for the whole fall: real Cd shifts a little with speed and posture, and real density grows as an object descends, so a long fall is approximated, not simulated. It ignores buoyancy, fair for dense objects in air but not for balloons, and it ignores wind, tumbling, and spin, which can change the area shown to the airflow second by second. Cd itself is empirical: the presets are reference values, not measurements of your object, and a Cd error of 20% moves the answer by about 10%. Treat the output as the right physics for the inputs you gave it, not a guarantee about a specific fall.

Frequently asked questions

Add This Calculator to Your Website

References