Terminal velocity calculator

Terminal velocity is the steady falling speed at which gravitational weight equals aerodynamic drag so net force and acceleration drop to zero. For bodies moving fast enough that inertia dominates viscosity, drag grows with the square of speed and the balance equation is mass times gravity equals one half times fluid density times velocity squared times the drag coefficient times the projected area. Solving for speed yields the closed-form square root of two m g over rho Cd A. Engineers, skydivers, drone designers, and students use that expression to size parachutes, estimate impact speeds, or check order-of-magnitude free-fall numbers without integrating the full differential equation of motion. This calculator takes mass, dimensionless drag coefficient, air density, cross-sectional area, and an optional gravity value, then returns terminal velocity plus weight, the matching drag force at that speed, and dynamic pressure. Non-positive area or density is rejected with a clear validation error instead of producing infinity or nonsense.

How quadratic-drag terminal velocity is derived

Start from Newton second law for a body falling under constant gravity with a speed-dependent drag force that opposes velocity. In the high Reynolds-number regime the drag magnitude is one half rho times v squared times Cd times A, where rho is fluid density, Cd is a dimensionless coefficient that folds shape and orientation, and A is the projected area normal to the flow. Weight is m times g.

While the body is still accelerating, weight exceeds drag and speed climbs; as speed rises, drag grows until the two forces match. Setting m g equal to one half rho v squared Cd A and solving for v produces v terminal equals the square root of two m g divided by rho Cd A. That algebraic step assumes density, Cd, area, and g are constant during the approach to steady state, which is the usual textbook and first-order engineering model.

Linear Stokes drag, by contrast, is proportional to speed rather than speed squared and applies to microscopic particles at very low Reynolds number; that problem is a different search intent and a different formula. Coefficients must be finite; NaN and Infinity are rejected. The special cases of zero or negative area and density are rejected with invalid input because they would divide by zero or invent non-physical densities, matching the product rule that non-positive area or density must fail loudly.

What you send and what you get back

Mass is the body mass in kilograms or any consistent mass unit. Drag coefficient is dimensionless; typical order-of-magnitude values include roughly zero point four seven for a smooth sphere and about one for a skydiver in a belly-to-earth posture. Air density is fluid density in kilograms per cubic metre; International Standard Atmosphere sea-level air is about one point two two five.

Cross-sectional area is the projected frontal area in square metres. Mass, drag coefficient, and g must also be positive. Wire the free browser form for homework and quick sanity checks, then call the identical contract from scripts when you need bulk tables for curriculum or design envelopes.

Where terminal velocity calculations show up in real work

Introductory physics and engineering mechanics courses assign free-fall with air resistance problems that ask for the asymptotic speed rather than the full time history; a single trusted oracle that returns the square-root closed form speeds grading and demo setup. Skydive and BASE-jump education materials quote belly-to-earth and freefly speeds that depend on mass, Cd, and area; instructors need reproducible numbers when students change body mass or canopy packing assumptions.

Unmanned aerial vehicle and model-rocket recovery design uses the same balance to size drogue and main parachutes so descent rate stays inside structural and soft-landing limits. Ballistics and sports science estimate impact speeds of raindrops, hail, or sports balls as a first filter before full CFD. Safety and drop-test planning compare predicted terminal speed against instrumented freefall data to catch unit mistakes early.

If you later need Stokes settling velocity for microparticles, time-dependent approach curves with numerical integration, altitude-varying density profiles, or horizontal projectile kinematics without drag, treat those as separate search intents and landings rather than overloading this terminal velocity calculator page.