EngineeringFormulaEngineering Mechanics
Terminal Velocity Calculator
Calculate terminal velocity from mass, drag coefficient, fluid density, and area, or solve for area from a target speed.
Terminal velocity is the constant speed a falling object reaches once drag force exactly balances gravity, so it stops accelerating. This calculator solves v = √(2mg / (Cd ρ A)) for terminal velocity given mass, drag coefficient, fluid density, and cross-sectional area - or, in the other direction, solves for the cross-sectional area needed to hit a target descent speed, the practical question behind sizing a parachute canopy or any descent-rate-limited design.
m · Mass of the falling object
Cd · Dimensionless - see reference table
ρ · 1.225 kg/m³ is air at sea level
A · Area perpendicular to the direction of fall
g · Optional - defaults to standard gravity (9.80665 m/s²)
Solution
Enter mass, drag coefficient, fluid density, and area (or a target speed) to calculate.
v = √(2mg / (Cd ρ A))
Formula Sheet
- vTerminal Velocity
- mMass
- CdDrag Coefficient
- ρFluid Density
- ACross-Sectional Area
- gGravity
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| v | Terminal Velocity | The constant falling speed reached once drag balances gravity. | m/s, mph, km/h |
| m | Mass | Mass of the falling object. | kg, lb |
| Cd | Drag Coefficient | Dimensionless shape factor describing how strongly the object resists motion through the fluid. | |
| ρ | Fluid Density | Density of the fluid the object falls through (commonly air). | kg/m³, lb/ft³ |
| A | Cross-Sectional Area | The object's area perpendicular to the direction of fall. | m², ft² |
| g | Gravity | Gravitational acceleration. | m/s², ft/s² |
How to Use This Calculator
- 01Choose a mode: solve for Terminal Velocity (given area), or solve for Area (given a target terminal velocity).
- 02Enter the object's mass and drag coefficient. Drag coefficient depends entirely on shape - see the reference table below for typical values.
- 03Enter the fluid density. Air at sea level (1.225 kg/m³) is the default; change it for a different altitude or a different fluid entirely.
- 04Enter cross-sectional area (velocity mode) or target terminal velocity (area mode).
- 05Gravity defaults to standard gravity (9.80665 m/s²) - override it only for a non-Earth or otherwise non-standard scenario.
How the Formula Works
As an object speeds up while falling, drag force grows with the square of velocity while weight stays constant, so the two forces eventually balance: 0.5 × Cd × ρ × A × v² = mg. Once that balance is reached, net force is zero and the object stops accelerating - that constant speed is terminal velocity.
Solving that balance equation for v gives v = √(2mg / (Cd ρ A)): a heavier object reaches a higher terminal velocity, while a larger area, higher drag coefficient, or denser fluid all reduce it, since each one increases drag force at a given speed.
Solving the same equation for A instead - A = 2mg / (Cd ρ v²) - answers a design question rather than a physics-homework one: given a target descent speed (a safe parachute landing speed, for example), how large does the drag-producing area need to be. Both directions use exactly the same underlying force balance.
Drag coefficient is a shape property, not a fixed physical constant - a flat plate, a sphere, and a streamlined body moving at the same speed through the same fluid experience very different drag purely because of their shape. The reference table below gives typical values, but a specific object's actual Cd depends on its exact geometry and surface finish.
Worked Example 01
Skydiver terminal velocity, belly-to-earth
Known
- Mass (m): 75 kg
- Drag Coefficient (Cd): 0.7
- Fluid Density (ρ): 1.225 kg/m³
- Cross-Sectional Area (A): 0.6 m²
Formula
v = √(2mg / (Cd ρ A))
Substitution
v = √(2 × 75 × 9.80665 / (0.7 × 1.225 × 0.6))
Result
v ≈ 53.5 m/s (≈192.5 km/h, ≈119.6 mph)
A 75 kg skydiver in a belly-to-earth position (Cd ≈ 0.7, area ≈ 0.6 m²) reaches a terminal velocity of about 53.5 m/s - close to the commonly cited real-world skydiving terminal speed of roughly 120 mph.
Worked Example 02
Parachute area for a target landing speed
Known
- Mass (m): 90 kg
- Drag Coefficient (Cd): 1.5
- Fluid Density (ρ): 1.225 kg/m³
- Target Terminal Velocity (v): 5 m/s
Formula
A = 2mg / (Cd ρ v²)
Substitution
A = 2 × 90 × 9.80665 / (1.5 × 1.225 × 5²)
Result
A ≈ 38.4 m²
For a 90 kg total load under a Cd ≈ 1.5 canopy, reaching a safe 5 m/s landing speed needs a canopy area of about 38.4 square meters - in the realistic range for round and square parachute canopy designs.
Applications
- 01Estimating a skydiver's or falling object's terminal velocity in a given body position or orientation
- 02Sizing a parachute canopy's area for a target, safe landing speed
- 03Comparing how much terminal velocity changes with body position, equipment, or altitude (air density)
Typical Drag Coefficients (Cd) by Shape
| Shape / Object | Approx. Cd |
|---|---|
| Streamlined airfoil | 0.045 |
| Sphere (smooth) | 0.47 |
| Skydiver, belly-to-earth | 0.6 – 1.0 |
| Skydiver, head-down | 0.15 – 0.3 |
| Parachute canopy (round/square) | 1.0 – 1.75 |
| Flat plate (perpendicular to flow) | 1.28 |
Assumptions
- 01Drag follows the standard quadratic drag equation (Fd = 0.5 Cd ρ A v²), valid for the turbulent, high-Reynolds-number regime typical of falling objects in air - not the linear (Stokes') drag regime that applies to very small or very slow-moving objects.
- 02Drag coefficient and cross-sectional area are treated as constant throughout the fall - a tumbling or reorienting object (a skydiver changing body position, for example) has a Cd and A that actually change over time.
- 03Fluid density is treated as uniform - a long fall through changing air density with altitude would need the calculation repeated at each altitude band, not a single average value.
Where This Model Stops
- 01Does not calculate the time or distance needed to reach terminal velocity - this calculator gives the final constant speed only, not the approach to it.
- 02Does not model non-uniform or turbulent flow effects beyond the single drag coefficient, such as vortex shedding or flow separation changes at different speeds.
- 03Drag coefficient values are typical references, not a substitute for wind-tunnel or CFD data for a specific object's actual geometry.
References
- [1]6.4 Drag Force and Terminal Speed
OpenStax University Physics Volume 1
Defines the quadratic drag equation and derives the terminal velocity formula used on this page.
- [2]Shape Effects on Drag
NASA Glenn Research Center
Reference for how drag coefficient varies by object shape, used for the typical-value table below.
Frequently Asked Questions
Why does a heavier object have a higher terminal velocity?
Because drag has to grow to match a larger weight before the forces balance, and since drag force grows with the square of velocity, a heavier object needs a higher speed to generate that larger drag force - all else (shape, area, fluid) held equal.
Does a bigger parachute always mean a slower, safer landing?
Yes, all else equal - terminal velocity is inversely proportional to the square root of area, so increasing area reduces landing speed, though not in direct proportion (doubling area reduces speed by a factor of about 1.41, not 2). Real canopy design also has to balance area against weight, packability, and opening-shock forces, not just terminal velocity alone.
Why is drag coefficient given as a single number instead of calculated from shape directly?
Because drag coefficient bundles up complex, shape-dependent flow behavior (boundary layer separation, wake turbulence) that isn't practical to derive from first principles for most real geometries. Cd values are determined experimentally (wind tunnel testing) or computationally (CFD) for specific shapes and then published as reference figures, which is what the table on this page provides.
Does terminal velocity depend on the height an object falls from?
No - terminal velocity itself depends only on mass, drag coefficient, area, fluid density, and gravity, not on drop height. Height determines whether the object has fallen far enough to actually reach terminal velocity in the first place, but the terminal velocity value itself is fixed once those other quantities are fixed.