Terminal Velocity Calculator - Free Fall Aerodynamic Speed Solver

The terminal velocity calculator determines the maximum, constant speed that a free-falling object reaches when the fluid resistance opposing its movement matches the downward gravitational force. This calculator makes this complex force balance simple by solving the quadratic drag equation, showing the exact speed at which an object stops accelerating and continues downward at a steady pace. This aerodynamic concept describes everything from raindrops falling from storm clouds to skydivers descending through the atmosphere.

• • Skydiver flight planning: Calculate how body position (belly-to-earth vs. head-down) changes drag coefficient and alters descent velocity.
• • Raindrop and hail speed analysis: Estimate the speed at which precipitation hits the ground based on droplet size and mass.
• • Aerosol and particle settling: Model the terminal settling velocities of fine dust or particulate matter in air or liquids.
• • Ballistics and atmospheric entry: Calculate the terminal speed of capsules, meteorites, or projectiles as they plunge through Earth's atmosphere.

When an object begins its descent, the only force acting on it is gravity, causing it to accelerate rapidly. As its downward speed increases, the upward drag force of the fluid (such as air) grows in proportion to the square of the velocity. Eventually, the upward drag force matches the downward weight of the object, bringing the net force to zero. With no net force, acceleration ceases entirely.

This state of zero acceleration is known as terminal velocity. The object continue descending at this constant speed until hitting the ground or changing shape. A skydiver who spreads their limbs increases their cross-sectional area, boosting drag and dropping their terminal speed. By pulling their limbs in, they reduce their area and descend significantly faster.

Before reaching steady speed, the object travels a certain free fall distance under changing acceleration.

The terminal velocity calculator processes variables using the standard drag equation to evaluate the balanced state where drag balances weight. By inputs in standard metric units, it evaluates the steady-state downward falling speed.

• v_t: Terminal velocity in metres per second (m/s).
• m: Mass of the falling object in kilograms (kg).
• g: Acceleration due to gravity in metres per second squared (m/s²).
• rho (rho): Density of the fluid in kilograms per cubic metre (kg/m³).
• A: Projected frontal cross-sectional area of the object in square metres (m²).
• C_d: Dimensionless drag coefficient modeling the aerodynamic slickness of the shape.

The calculator processes inputs in standard metric units. It first multiplies the mass by gravity and by two to find the weight-driven force component. Next, it computes the aerodynamic resistance factor by multiplying the fluid density, projected cross-sectional area, and drag coefficient together. Finally, it divides the weight factor by the drag factor and extracts the square root.

If any input is zero or negative, the calculator rejects the calculation because a physical object must possess positive mass, area, drag, density, and gravitational pull to establish a valid terminal speed.

According to Encyclopedia Britannica, terminal velocity is the constant speed at which a falling body travels when the resistance of the medium through which it is falling prevents further acceleration.

The math behind this balance relies on the standard drag equation to compute the upward force.

Understanding terminal velocity requires breaking down the physics of fluids and gravity into four core components.

These concepts explain why two objects of identical mass can fall at wildly different speeds if their shapes or orientations differ. The relationship between these variables determines the behavior of objects moving through any fluid medium.

The terminal velocity calculator solves for speed, mass, area, fluid density, or gravity in five easy steps. Use the default settings to explore gravity-drag balance.

1. 1 Enter the object mass: Provide the mass of the falling body in kilograms. You can convert pounds to kilograms by dividing the weight by 2.2046.
2. 2 Specify the gravity: Input the gravitational acceleration. The default is Earth's surface gravity (9.80665 m/s²), but you can input values for other planets or altitudes.
3. 3 Set the fluid density: Enter the density of the medium (usually air). Sea-level air at room temperature is approximately 1.204 kg/m³, but it drops at high altitude.
4. 4 Input the projected area: Enter the projected frontal surface area in square metres. For humans, this ranges from 0.18 m² (head-first) to 0.7 m² (belly-to-earth).
5. 5 Set the drag coefficient: Input the dimensionless Cd value. Use 1.0 for a belly-to-earth skydiver, 0.47 for a sphere, or 1.15 for a flat circular disk.

Plugging in 80 kg mass, Earth gravity, standard air density, 0.7 m² area, and a drag coefficient of 1.0 yields a terminal velocity of 43.15 m/s. If the skydiver changes to a streamlined dive position, dropping their area to 0.18 m² and Cd to 0.7, the calculated speed climbs to 113.7 m/s.

To find out how long the initial acceleration phase takes, you can refer to the companion free fall time calculator.

Using the terminal velocity calculator offers key advantages over trying to solve fluid drag quadratic equations by hand.

• • Zero-error algebraic evaluations: Avoid mistakes with square roots and units by letting the script handle the mathematical order of operations.
• • Saves engineering design time: Quickly estimate terminal settling rates for industrial separation columns or atmospheric deceleration payloads.
• • Instant unit harmonization: Maintains correct relationships between mass, area, gravity, and fluid density without manual factor tracking.
• • Facilitates scientific experimentation: Compare theoretical terminal speeds with measured drop-tower experiment speeds to calculate experimental drag coefficients.
• • High precision rounding: Calculates results to 4 significant figures, meeting academic and engineering documentation standards.
• • Integrates with physics curriculum: Serves as an interactive companion tool for fluid dynamics, kinematics, and introductory physics courses.

At terminal speed, net acceleration drops to zero, which can be cross-verified with the acceleration calculator.

When using the terminal velocity calculator, several environmental and physical factors can alter the actual speed achieved in real life.

• • The calculator assumes a constant fluid density. In high-altitude jumps (like Felix Baumgartner's supersonic leap), air density changes dynamically by orders of magnitude as the object descends.
• • It treats the drag coefficient as a constant. In reality, Cd varies with velocity and Reynolds number, especially when approaching the speed of sound where compressibility effects occur.

For most practical, sea-level situations involving human-sized objects, this quadratic drag model provides highly accurate predictions.

For extremely small particles (such as fog droplets) or highly viscous liquids (such as honey), you should use Stokes' Law instead of this quadratic model.

According to Wikipedia (Terminal Velocity), terminal velocity represents the steady-state speed where gravity and air resistance balance, depending heavily on the cross-sectional area, drag coefficient, and fluid density.

To calculate the density input at different temperature and altitude conditions, the air density calculator provides the required values.