Terminal Velocity Calculator

Introduction

Terminal velocity is the constant speed a falling object eventually reaches when the upward drag of the fluid it travels through exactly balances its downward weight. Nothing dramatic happens at that instant: the forces simply add up to zero, so the object stops accelerating and continues at a steady speed for the rest of the fall. Drag grows with the square of speed, which means a slowly moving object barely feels the air while a fast one is pushed back hard. Somewhere between those two extremes the drag force catches up with the weight, and that meeting point defines the terminal velocity [openstax-drag].

This balance shows up everywhere. A skydiver in a stable belly-to-earth position levels off at roughly 43 m/s (about 154 km/h) and would keep falling at that speed indefinitely if the parachute never opened. Raindrops arrive at only a few meters per second, which is why a cloud can dump water on you all afternoon without leaving a bruise, while a hailstone of similar size hits hard enough to dent a car roof. Sediment settles slowly at the bottom of a lake because its terminal velocity in water is tiny, and weather balloons rise at a predictable, nearly constant rate for exactly the same reason — with the buoyant force playing the role that weight plays for a falling object. Drag and terminal speed also set the working envelope of every parachute, drone recovery system, and atmospheric probe ever designed.

The Terminal Velocity Calculator solves the force balance for you. Enter the object's mass, its drag coefficient, its cross-sectional area, the density of the fluid it moves through, and the local gravitational acceleration, and the tool returns the terminal speed in meters per second and kilometers per hour, plus the drag force acting at that speed. If you also enter the volume of fluid the object displaces, the calculator adds the buoyant force and subtracts it from the weight, which is what you need for balloons, airships, and anything falling through water rather than air.

The result connects directly to the rest of the mechanics toolkit on this site. The Force Calculator supplies the weight side of the balance through F = m g, the Acceleration Calculator shows how quickly the object approaches terminal speed, and the Projectile Motion Calculator handles the horizontal component of trajectories where drag matters. When you need the energy the falling body carries rather than its speed, the Kinetic & Potential Energy Calculator converts the terminal velocity into joules. [nasa-termv]

How to Use

Enter the six input values and the results update immediately. Nothing needs to be converted: mass is in kilograms, area in square meters, densities in kilograms per cubic meter, and gravity in meters per second squared.

1. Mass (kg) — the object's mass, not its weight. For an 80 kg person, enter 80.
2. Drag Coefficient — a dimensionless number that describes how streamlined the shape is. The default of 1.0 fits a person in a belly-to-earth skydiving position and is a reasonable starting guess for blunt bodies. A smooth sphere is closer to 0.47, and an open parachute canopy reaches about 1.5.
3. Cross-Sectional Area (m²) — the area the object presents to the oncoming flow, measured perpendicular to the direction of motion. A skydiver in a stable spread position presents about 0.7 m²; head-down, it drops to roughly 0.2 m².
4. Fluid Density (kg/m³) — 1.225 for air at sea level and 15 °C, which is the default. Water is roughly 1000, and air at 10,000 m is only about 0.41.
5. Gravity (m/s²) — 9.81 on Earth at sea level, 1.62 on the Moon, 24.79 on Jupiter.
6. Displaced Volume (m³, optional) — the volume of fluid the object pushes aside as it moves. Leave it at zero for a dense object in air; fill it in for balloons, submarines, or anything whose volume is not negligible compared to its mass.

The Formula

Terminal velocity comes from a single idea: at the terminal speed the drag force equals the weight of the object, so the net force is zero. The drag force on a blunt body moving quickly through a fluid is given by the drag equation [openstax-drag]:

where rho is the fluid density, A is the cross-sectional area, C_d is the drag coefficient, and v is the speed. Setting this equal to the weight m g gives the equilibrium condition that defines terminal velocity [nasa-termv]:

Solving for the unknown speed means undoing a square, which is why the result is a square root rather than a simple ratio. Multiply both sides by two, divide by the drag bracket, and take the square root:

The square root is the whole story of the scaling behaviour. Doubling the mass raises the terminal speed by a factor of 1.414, not by two, and doubling the area, density, or drag coefficient lowers it by 0.707, not by half. Each variable acts inside the root, so the object's response is always damped by comparison with the naive expectation.

Two consequences are worth spelling out. First, the height of the fall does not appear anywhere in the equation. Drop a stone from 50 m or from 500 m through air of the same density and the terminal speed is identical; what changes is how long the fall takes and how much of it is spent accelerating. (Altitude matters for a different reason — thinner air, handled by the density input — which the next section covers.) The speed approaches its limit asymptotically, following a hyperbolic tangent curve [hyperphysics-airfric]:

For the skydiver of Example 1 this means roughly eight seconds and about 220 m of fall to reach 95 percent of the terminal speed, with the last few percent taking much longer. Second, gravity enters as the square root, not linearly. Going from Earth to the Moon would multiply the terminal velocity by the square root of 1.62/9.81, or about 0.41 — but the Moon has almost no atmosphere, so the question never arises in practice; the fall is simply unlimited free fall.

Reference Table: Drag Coefficients and Terminal Speeds

The drag coefficient is the least intuitive input in the calculation, because it cannot be computed from first principles for a real object — it is measured in wind tunnels and water channels and then tabulated. The values below are representative of blunt and streamlined bodies at Reynolds numbers around 10⁴ to 10⁶, the range that covers falling people, sports balls, and parachutes [wiki-dragco]. The final column shows the terminal velocity of an 80 kg body with a 0.7 m² frontal area in sea-level air, so the effect of each coefficient is visible directly.

The chart shows why streamlining has diminishing returns. Moving from a blunt plate to a sphere cuts the coefficient by more than half but changes the terminal speed by only about 65 percent, because the square root compresses every gain in coefficient into a smaller gain in speed. Meanwhile the parachute entry in the table is misleading unless the area is considered too: a real canopy of 50 m² with a coefficient of 1.5 brings the same 80 kg person down at about 4.1 m/s rather than 34.93 m/s, because area, not the coefficient alone, is what makes a parachute work.

Area is also the reason the reference area must be stated whenever a coefficient is quoted. A number measured against frontal area and one measured against wetted surface area differ by an order of magnitude for the same body, and mixing them silently corrupts the calculation [nasa-dragco].

Practical Tips

Match the coefficient to the reference area. A drag coefficient has no meaning on its own. If the value was measured against frontal area, the area you enter must also be frontal. This single mismatch is the most common way to get a terminal velocity that is off by a factor of two or more [nasa-dragco].

Pick a coefficient for the right speed regime. A coefficient measured at low speed cannot be reused at high Mach number, because viscosity and compressibility effects change the flow. NASA notes explicitly that applying a low-speed coefficient at twice the speed of sound is incorrect. For falls through air below a few hundred kilometers per hour, standard tabulated values are appropriate.

Remember that the drag output is a built-in check. At terminal velocity the drag force equals the effective weight, so the calculator's drag result should match m g whenever the displaced volume is zero. In Example 1 that is 784.80 N for 80 kg. If the two disagree, check whether a volume was entered by mistake.

Use mass, not weight. The input is kilograms. If you are working from a force in pounds-force or a scale reading in pounds, convert to kilograms first — entering a weight in pounds as a mass in kilograms inflates the mass by a factor of 2.2 and therefore the terminal speed by about 1.48.

Expect the approach to be gradual. Terminal velocity is a limit, not a switch. A skydiver needs roughly eight seconds and about 220 m of fall to get within 5 percent of it, and the remaining gap closes slowly [hyperphysics-airfric]. Short falls never reach terminal velocity at all, which is why dropping a ball from a table is a poor experiment for measuring it.

Choose the fluid deliberately. The default density is sea-level air at 1.225 kg/m³. Water is about 800 times denser, so for the same mass, area, and drag coefficient the terminal velocity in water is smaller by a factor of about 28. Entering water's density without also entering the displaced volume ignores the buoyant force, and for a body that is not much denser than water that omission dominates the result.

Do not forget buoyancy for anything that floats or nearly floats. A weather balloon, a submarine, a drifting swimmer, and a sediment particle in water all depend on the apparent weight rather than the full weight. The buoyant force output tells you how much of the weight is being cancelled before the object has even started to move [hyperphysics-buoy].

Use the result to design rather than to predict only. Because area sits inside the square root, a canopy four times larger roughly halves the descent speed. That single relationship is how parachute sizing works, and it is the reason a reserve canopy and a main canopy can be compared numerically before anything is packed.

Limitations

Constant drag coefficient. The model treats C_d as a fixed number, but real coefficients drift with speed, turbulence, and orientation. A tumbling object changes shape and coefficient continuously, a parachute in the process of opening sweeps through a wide range of effective areas, and a body in an unstable position can oscillate between configurations. The calculated value is best understood as the terminal speed for one specific, stable attitude.

Linear drag is out of scope. Everything above assumes quadratic drag, which is valid at high Reynolds numbers for large, fast-moving bodies. For particles that are very small or very slow, drag becomes proportional to speed and the terminal velocity follows the Stokes relation instead. Using the square-law version for a micron-scale particle gives a badly wrong answer [openstax-drag].

Constant fluid density. Real falls cross strata of changing density. A fall from 10,000 m starts in air at about 0.41 kg/m³ and ends in air at 1.225 kg/m³, so the true terminal velocity is not one number but a moving target that decreases as the object descends [nasa-atmos]. The calculator gives the terminal velocity for the density you enter, which is the right answer only while the object stays in that layer.

No horizontal motion, wind, or turbulence. The balance is written along the vertical axis alone. Crosswinds, updrafts, thermals, and the lateral drift of a falling body are not represented. A body in a strong updraft can have a downward terminal velocity relative to the air and still be rising relative to the ground.

No rotation or Magnus effect. A spinning ball or a curved parachute generates lift-like forces perpendicular to the flow that change the trajectory. The model ignores both the spin and the resulting side force.

No compressibility or rarefied-flow corrections. At speeds approaching the local speed of sound, shock waves add wave drag and the coefficient changes. In near-vacuum conditions at extreme altitude, the continuum assumptions behind the drag equation break down entirely, and rarefied-gas models are required instead.

Zero terminal velocity is not predicted. If an entered volume makes the buoyant force equal the weight, the equation has no solution. The calculator returns no result rather than reporting a hovering state, and negative apparent weight — a rising object — is outside the tool's intended use.

Rigid body assumption. The object is treated as a single rigid body with a fixed mass distribution. Elastic deformation, fluttering fabric, and breakup are not modelled, so the estimate should be treated as an upper or lower bound for objects that change shape as they fall.

Frequently Asked Questions

❓ What is terminal velocity in simple terms?

✅ It is the steady speed a falling object settles into when the air pushing up on it balances its weight pulling it down. Drag grows with the square of speed, so the faster the object goes, the harder the air resists. At some speed the two match, the net force becomes zero, and the object stops speeding up. For an 80 kg skydiver in a belly-to-earth position that speed is about 42.8 m/s, or 154 km/h.

❓ Does terminal velocity depend on the height of the fall?

✅ No. The height of the fall does not appear anywhere in the equation. Drop the same object from 500 m or from 5,000 m through air of the same density and the terminal speed is identical; what changes is how much of the fall is spent accelerating and how long the steady-speed phase lasts. Air density itself does change with height, and thinner air produces a faster terminal speed, but that is an effect of the fluid, not of the drop height.

❓ Does a heavier object have a higher terminal velocity?

✅ Yes, for objects of the same size and shape, but only in proportion to the square root of the mass. Doubling the mass increases the terminal speed by 41 percent, not by 100 percent. That is why a heavy equipment pack changes a skydiver's free-fall speed much less than its effect on the total weight would suggest.

❓ Why does the formula have a square root in it?

✅ Because drag depends on the square of speed. The balance of forces produces an equation where the unknown speed is squared, and undoing a square means taking a square root. That is also why every input has a damped effect: doubling any term under the root changes the answer by a factor of about 1.41 rather than by two.

❓ Why is the drag force at terminal velocity exactly equal to the weight?

✅ Because that equality defines terminal velocity. If drag were smaller than the weight, the object would still be accelerating and would speed up. If drag were larger, the object would be slowing down. Only when the two are equal is the net force zero and the speed constant, which is why the calculator reports 784.80 N of drag for an 80 kg object at terminal velocity.

❓ What drag coefficient should I use for a person?

✅ About 1.0 to 1.3 for a person in a stable belly-to-earth position, roughly 0.7 for a feet-first or a tight head-down position, and around 0.3 to 0.5 for a well-streamlined posture with arms at the sides. The default of 1.0 in the calculator is a good starting point for the standard skydiving position. Note that any published coefficient must be paired with the same reference area it was measured against.

❓ How long does it take to reach terminal velocity?

✅ Longer than most people assume. A skydiver reaches about 95 percent of terminal velocity after roughly eight seconds and about 220 m of fall, and closing the remaining five percent takes considerably longer because the approach is asymptotic. Short drops, such as anything from a tabletop or a typical balcony, never come close to terminal velocity.

❓ What is the difference between terminal velocity and free fall?

✅ Free fall usually means motion under gravity alone, with no drag, in which case the object keeps accelerating indefinitely. Terminal velocity is what happens when drag is included: the object accelerates only until drag cancels its weight, then falls at a constant speed. In a vacuum there is no terminal velocity at all, which is why lunar missions can drop objects from any height without a limit speed.

❓ Why does a parachute slow a fall so dramatically?

✅ Mostly because of area. A canopy about 50 m² with a drag coefficient of 1.5 brings an 80 kg person down at roughly 4.1 m/s, compared with 42.8 m/s in free fall. Because area sits inside the square root, a canopy four times larger halves the descent speed, so parachute design is largely a question of how much area can be deployed reliably.

❓ Does terminal velocity apply to objects that rise?

✅ Yes, with the driving force reversed. A helium balloon, a weather balloon, or any body whose displaced fluid weighs more than the body itself has a negative apparent weight, so it rises and reaches a constant rise speed governed by the same balance. The calculator handles this through the displaced volume input, which subtracts the buoyant force from the weight.

❓ Why do raindrops not hurt when they hit you?

✅ Because a raindrop is small and light, with a lot of frontal area for its mass, so the drag balance is reached at a low speed. A drop about 2 mm across settles at roughly 9 m/s, fast enough to soak you and far too slow to injure. Cloud droplets are a thousand times smaller still and drift almost motionless, because at that scale drag becomes proportional to speed rather than to speed squared. Hailstones of a similar size are far denser, reach much higher terminal speeds, and carry considerably more kinetic energy on impact.

❓ Does terminal velocity change with altitude?

✅ Yes, substantially. Air density at 10,000 m is about 0.41 kg/m3, roughly a third of the sea-level value, and terminal velocity varies with the inverse square root of density. An 80 kg skydiver therefore falls at about 73.6 m/s at 10,000 m and only 42.8 m/s at sea level. Because the object descends through denser air, the real fall has no single terminal velocity at all.