Terminal Velocity Calculatorvt = √(2mg ÷ ρACd)
Terminal velocity is the steady speed a falling object reaches when air drag grows to equal its weight: vt = √(2mg ÷ ρACd). A belly-down skydiver levels off near 50 m/s (180 km/h). Enter a mass, drag coefficient and frontal area, or the size and density of a ball, raindrop or hailstone. You also get how long and how far it takes to reach 95% of top speed.
Formula verified against NASA Glenn — Beginner’s Guide to Aeronautics · OpenStax — University Physics, Volume 1
By Manzoor Ahmad · Pharmacist & calculator developerReviewed & updated
Falling object
Sea-level air is 1.225 kg/m³. Thinner air at altitude raises terminal velocity.
Step-by-step working
What is terminal velocity?
Terminal velocity is the constant speed a freely falling object eventually reaches when the drag force from the surrounding air equals the object’s weight. At that point the net force is zero, so the object stops accelerating.
It explains why raindrops don’t hit like bullets, why skydivers can change their fall rate by changing body position, and why parachutes work: they multiply the area A and drag coefficient Cd.
Quick answers
What is the terminal velocity of a human?
About 53 m/s (195 km/h, 120 mph) for a belly-to-earth skydiver, and 80–90 m/s head-down. Wingsuits slow the descent to roughly 20–30 m/s vertically by generating lift.
How fast does a raindrop fall?
A typical 2 mm drop falls at about 6.5 m/s (23 km/h). The biggest drops, around 5 mm, top out near 9 m/s because they flatten and then break up. Load the raindrop preset to check.
Does a heavier object have a higher terminal velocity?
For the same shape and size, yes: vt scales with √m. That is why a steel ball overtakes a ping-pong ball of the same size, even though in a vacuum they would fall together (see the free fall calculator).
How long does it take to reach terminal velocity?
From rest you hit about 95% of vt after t ≈ 1.83·vt ÷ g. For a skydiver that is roughly 9–10 s; for a raindrop, well under a second.
Terminal velocity formula
Assumes quadratic (turbulent) drag, which suits people, balls and raindrops larger than about 1 mm.
| Symbol | Meaning | SI unit |
|---|---|---|
| vt | Terminal velocity | m/s |
| m | Mass | kg |
| g | Gravitational acceleration | m/s² |
| ρ | Air (fluid) density | kg/m³ |
| A | Frontal (projected) area | m² |
| Cd | Drag coefficient | — |
How to calculate terminal velocity
- Find the weight: mg, in newtons.
- Estimate the drag setup: frontal area A and drag coefficient Cd for the shape and orientation.
- Set drag equal to weight: ½ρv²ACd = mg.
- Solve for v: vt = √(2mg ÷ ρACd).
Worked example: a skydiver in the belly-to-earth position
An 80 kg skydiver (with kit) falls belly-down with Cd ≈ 1.0 and a frontal area of about 0.5 m², in sea-level air.
vt = √2,562 = 50.6 m/s ≈ 182 km/h ≈ 113 mph
She reaches 95% of that speed after about 9.5 s and roughly 300 m of fall. Tucking into a head-down dive (Cd ≈ 0.7, A ≈ 0.25 m²) pushes vt to about 86 m/s, more than 300 km/h.
Typical terminal velocities
Measured or well-established values in sea-level air.
| Object | Terminal velocity | km/h |
|---|---|---|
| Skydiver, belly-down | ≈ 50–55 m/s | 180–200 |
| Skydiver, head-down | ≈ 80–90 m/s | 290–320 |
| Raindrop, 2 mm | ≈ 6.5 m/s | 23 |
| Large raindrop, 5 mm | ≈ 9 m/s | 32 |
| Hailstone, 3 cm | ≈ 25 m/s | 90 |
| Baseball | ≈ 33–42 m/s | 120–150 |
| Cat (spread out) | ≈ 27 m/s | ≈ 100 |
Common mistakes to avoid
- Using a cross-section that doesn’t face the airflow. A is the area the air “sees”, not the total surface area.
- Assuming terminal velocity is reached instantly. A skydiver needs about 10 s and 300 m to get close.
- Ignoring altitude. At 4,000 m the air density is about 0.82 kg/m³, so vt is roughly 22% higher than at sea level.
Manzoor Ahmad
Pharmacist turned Web Developer & Calculator Engineer · 13+ years in web developmentI’m a pharmacist by training who turned into a full-time tech geek, and for 13+ years I’ve designed and coded web tools. My pharmacy degree included physical pharmacy and pharmaceutics — states of matter, heat, flow and reaction kinetics — so the physics behind these tools is familiar ground, not a new hobby.
I’m not a physicist, so I don’t ask you to take my word for it: every formula on this page is traced to an open university textbook or NIST reference data, and the calculator is tested against published worked examples before it goes live. Spot an error? Tell me and it will be fixed and dated on the next update.
Terminal Velocity Calculator: frequently asked questions
vt = √(2mg ÷ ρACd), where m is mass, g gravity, ρ the fluid density, A the frontal area and Cd the drag coefficient.
It’s the maximum steady speed a falling object reaches, the point where air resistance pushes up as hard as gravity pulls down, so the object stops accelerating.
No. It depends on mass, shape, size and the density of the air or fluid. A feather and a bowling ball have very different terminal velocities.
Yes. Thinner air at altitude means less drag, so terminal velocity rises. Felix Baumgartner broke the sound barrier in 2012 by jumping from about 39 km, where the air is extremely thin.
Impact at terminal velocity is usually fatal without a parachute. There are rare recorded survivals, often involving snow, trees or other energy-absorbing surfaces.
Use about 1.0 for a belly-down skydiver, 0.47 for a smooth sphere, 0.3 for a baseball, and 0.25–0.35 for a car. Values change with speed and surface roughness.
References & sources
- NASA Glenn — Beginner’s Guide to Aeronautics — terminal velocity and the drag equation
- OpenStax — University Physics, Volume 1 — Ch. 6.4, Drag Force and Terminal Speed
- HyperPhysics, Georgia State University
Disclaimer: This calculator is an educational tool built on idealised textbook models. It is not a substitute for professional engineering or safety calculations.