Terminal Velocity Calculator gives the steady speed a falling object settles at, the time and distance it takes to get within 1% of the limit, and the drag force acting on it then.
Area is the outline the object shows to the air as it falls.
Air or Fluid, Gravity and Units
Step-by-Step Working
- Known values in SI units: m = 85 kg, A = 0.7 m², Cd = 1, ρ = 1.2 kg/m³, g = 9.81 m/s²
- At terminal velocity, drag equals weight: ½ × ρ × Cd × A × vₜ² = m × g
- Rearranged for the unknown: vₜ = √(2 × m × g ÷ (ρ × A × Cd))
- Weight side: 2 × m × g = 1,667.7 N
- Drag side: ρ × A × Cd = 0.8429 kg/m
- Divide and take the square root: vₜ = √1,978.6 = 44.48 m/s
- Time to 99% from rest: (vₜ ÷ g) × tanh⁻¹(0.99) = 12 s
- Result: vₜ = 44.48 m/s (160.13 km/h, 99.5 mph)
Terminal Velocity at Altitude
| Air or Fluid | Density | Terminal Velocity |
|---|---|---|
| Sea level, 15 °C | 1.225 kg/m³ | 44.1 m/s |
| Sea level, 20 °C | 1.2041 kg/m³ | 44.48 m/s |
| 1,000 m | 1.112 kg/m³ | 46.29 m/s |
| 2,000 m | 1.007 kg/m³ | 48.64 m/s |
| 3,000 m | 0.9093 kg/m³ | 51.19 m/s |
| 5,000 m | 0.7364 kg/m³ | 56.88 m/s |
| 10,000 m | 0.4135 kg/m³ | 75.91 m/s |
Speed-Time Graph
What the Terminal Velocity Calculator Works Out
The Terminal Velocity Calculator finds the steady speed a falling object settles at once air drag has grown to match its weight. You enter the mass, the area facing the air and a drag coefficient for the shape. Air density and gravity are preset and can be changed.
It also runs the same balance backward. Pick a target speed and it returns the mass or the area that produces it, or enter a speed you measured and it returns the drag coefficient. Each result comes with the time and distance needed to get there and a table for thinner air at altitude.
Terminal Velocity Formula and Variables
A falling object feels its weight pulling down and drag pushing up. Drag grows with the square of the speed, so the object speeds up until the two forces are equal. Setting drag equal to weight and solving for speed gives the terminal velocity formula.
$$v_t = \sqrt{\frac{2 m g}{\rho A C_d}}$$
Here m is the mass in kg, g is gravity in m/s², ρ is the density of the air or fluid in kg/m³, A is the cross-sectional area facing the flow in m², and Cd is the drag coefficient, which has no unit. The result vt comes out in m/s, and the first card converts it to km/h, mph and ft/s.
The same equation appears in OpenStax College Physics, which builds it from the drag force on large objects moving fast through air. It leaves out buoyancy, so results in water are only a rough guide. It also does not suit dust or droplets, where drag follows speed and not speed squared.
Terminal Velocity of a Human
Body position changes the answer more than body weight does. The two skydiver presets use the figures from the OpenStax textbook, with air at 20 °C and g = 9.81 m/s². The Terminal Velocity Calculator returns the values in this table.
| Position | Mass, area, Cd | Terminal velocity | Time to 99% | Distance to 99% |
|---|---|---|---|---|
| Belly to earth | 85 kg, 0.7 m², 1.0 | 44.48 m/s (160.13 km/h, 99.5 mph) | 12 s | 395 m (1,296 ft) |
| Head first | 75 kg, 0.18 m², 0.7 | 98.48 m/s (354.54 km/h, 220.3 mph) | 26.57 s | 1,936 m (6,353 ft) |
The textbook also quotes about 200 km/h for a spread-eagle skydiver, which is faster than the 160 km/h its own 0.7 m² example gives. The area estimate explains the gap. Switch the calculator to Area for a Target Speed, enter 200 km/h for the same 85 kg jumper, and it returns 0.449 m².
The head-first figure is more than twice the belly-down one, even though that skydiver is 10 kg lighter. A quarter of the area and a lower drag coefficient outweigh the loss of mass. This is why skydivers change speed in the air simply by changing shape.
Time to Reach Terminal Velocity
An object dropped from rest never reaches terminal velocity exactly. It closes in on it, quickly at first and then very slowly. With drag that grows as speed squared, the speed after a time t follows a hyperbolic tangent curve.
$$v(t) = v_t \tanh\left(\frac{g t}{v_t}\right)$$
The Terminal Velocity Calculator reports the time to reach 99% of the limit, which works out to a fixed multiple of vt divided by g. The Speed-Time Graph panel draws the same curve and marks that point with a dot.
$$t_{99} = 2.647 \frac{v_t}{g}$$
The belly-down skydiver needs 12 seconds and falls 395 m in that time. Half of the final speed arrives after only 2.49 seconds. At roughly 3 m per story, 395 m is about 130 floors, so a fall from an ordinary building ends well before terminal velocity.
Faster objects take longer. The head-first skydiver needs 26.57 seconds and 1,936 m to reach 99%. For a full fall from a set height, with speed and distance at each moment, use the Free Fall with Air Resistance Calculator.
Drag Coefficient, Sphere, Cylinder and Car
The drag coefficient is the hardest input to know, so the shape field carries presets. The two skydiver values and the Toyota Camry value of 0.28 match the OpenStax table. The sphere, hemisphere, cube and angled cube values match a wind-tunnel drag chart reproduced by MIT OpenCourseWare.
For a sphere, choose the circle unit in the area field and type the diameter. A 1 kg sphere 10 cm across, at the preset 0.47, has a terminal velocity of 66.44 m/s. OpenStax lists 0.45 for a sphere, which raises the result by 2%, to 67.9 m/s.
The long cylinder preset of 0.82 is for a cylinder falling end-first, with its flat face to the air. For any other shape, switch the Terminal Velocity Calculator to Drag Coefficient mode. Enter a terminal velocity you have measured, with the mass and area, and it solves for Cd.
The terminal velocity of a car depends on which way it falls. The 0.28 figure is for air meeting the car nose-first. With example inputs of 1,500 kg and 2.2 m² of frontal area, that gives 199.19 m/s, or 446 mph. A car falling flat shows far more area and would be much slower.
Balls are simpler. The baseball preset, 148.83 g and 9.25 inches around, gives 41.06 m/s (91.84 mph). A golf ball at the limits in the R&A equipment rules, 45.93 g and 42.67 mm, gives 36.68 m/s (82.05 mph) with the preset coefficient of 0.389.
Thinner Air Means a Faster Fall
Air density sits under the square root, so terminal velocity rises as the air thins. The altitude table in the Terminal Velocity Calculator reruns your object through standard-atmosphere densities. These are the values for the belly-down skydiver.
| Altitude | Air density | Terminal velocity |
|---|---|---|
| Sea level, 15 °C | 1.225 kg/m³ | 44.1 m/s (158.8 km/h) |
| Sea level, 20 °C | 1.2041 kg/m³ | 44.48 m/s (160.1 km/h) |
| 1,000 m | 1.112 kg/m³ | 46.29 m/s (166.6 km/h) |
| 3,000 m | 0.9093 kg/m³ | 51.19 m/s (184.3 km/h) |
| 5,000 m | 0.7364 kg/m³ | 56.88 m/s (204.8 km/h) |
| 10,000 m | 0.4135 kg/m³ | 75.91 m/s (273.3 km/h) |
This is also how something can fall faster than its sea-level terminal velocity. Felix Baumgartner reached 1,357.6 km/h, Mach 1.25, after leaving a balloon at 38,969 m in 2012, a record ratified by the FAI. The air up there is too thin to hold a falling body to ordinary speeds.
An object that enters thicker air above its local terminal velocity slows down until drag and weight balance again. The same happens to anything thrown downward faster than its limit. Terminal velocity is the speed a fall settles at, and the start can be above it or below it.
Terminal Velocity Facts and Figures
Is terminal velocity 9.8 m/s?
No. The 9.8 figure is the acceleration of gravity, 9.8 m/s², which is how fast speed builds at the start of a fall. Terminal velocity is a speed in m/s, and it is different for every object. At terminal velocity the acceleration has dropped to zero, because drag cancels the weight.
Does weight affect terminal velocity?
Yes. Terminal velocity rises with the square root of mass, so doubling the mass at the same size and shape raises it by 41%. The Terminal Velocity Calculator shows this on its last card. The 85 kg skydiver at 44.48 m/s would fall at 62.91 m/s with twice the mass, and at 31.45 m/s with twice the area.
The same card gives mass per drag area, 121.43 kg/m² for that skydiver. Objects with equal values fall at equal speeds in the same air. For bullets this ratio is the basis of the figure worked out by the Ballistic Coefficient Calculator.
How far would a human fall in 7 seconds?
About 181 m for the belly-down skydiver, who is doing 40.6 m/s by then, already 91% of terminal velocity. With no air the same 7 seconds would cover 240 m. Drag has little effect early on. After 2 seconds the fall is 62.4 ft with air and 64.4 ft without.
How long does it take to fall from 30,000 feet?
Roughly three and a half minutes belly-down, if the air had sea-level density all the way. That figure is 9,144 m at 44.48 m/s, plus a few seconds to get up to speed, or about 209 seconds. Real air is thinner near the top, where the table above shows 75.91 m/s at 10,000 m, so the true time is shorter.
Is terminal velocity faster than a bullet?
Not for a person. The head-first skydiver’s 98.48 m/s is under a third of the speed of sound, 343 m/s, and rifle bullets usually leave the barrel faster than sound. The Muzzle Velocity Calculator gives bullet speeds from energy and weight for a direct comparison.
Can a human survive a fall at terminal velocity?
The Terminal Velocity Calculator can only supply the numbers. At 44.48 m/s the skydiver carries 84.09 kJ, and hits as hard as after a 100.85 m drop with no air. A parachute works by adding area. Holding the same 85 kg jumper to 6 m/s takes 38.47 m² at a drag coefficient of 1.
Small animals fare better because their terminal velocity is low. OpenStax notes that a squirrel can fall from a significant height without injury for this reason. Less mass for each unit of area means a slower fall and a far gentler landing.
Does the Moon have a terminal velocity?
No. The Moon has no air to push back, so nothing balances the weight and a falling object keeps gaining speed until it lands. The Moon option in the Terminal Velocity Calculator changes g to 1.6 m/s² only. A result still needs a fluid density, so it describes Moon gravity acting in the air you entered.