Free Fall with Air Resistance Calculator

An object falling through air experiences gravity downward and aerodynamic drag upward. Drag increases with speed, so acceleration is largest at release and gradually decreases as the object speeds up. This calculator models quadratic drag, the common high-Reynolds-number approximation in which drag force equals one half times air density, drag coefficient, cross-sectional area, and speed squared. It assumes the object starts from rest, keeps a constant orientation and area, and falls through air of constant density under Earth gravity. Terminal velocity occurs when drag balances weight and net acceleration reaches zero. The calculator finds it from the square root of two times mass times gravity divided by air density, drag coefficient, and area. Velocity after an entered time uses the hyperbolic tangent solution. Distance uses the corresponding logarithm of the hyperbolic cosine solution. These closed-form equations avoid time stepping while preserving the gradual approach to terminal speed. Velocity approaches the terminal value asymptotically rather than crossing it. Mass, projected area, and drag coefficient must describe the same object and orientation. Cross-sectional area is the frontal area perpendicular to motion, measured in square meters. Drag coefficient is dimensionless and depends on shape, surface, airflow regime, and orientation. A sphere may use a value near 0.47 under suitable conditions, while a broad human posture can be higher. Standard sea-level air density is often approximated as 1.225 kilograms per cubic meter, but density decreases with altitude and changes with temperature, pressure, and humidity. This model is useful for comparing skydivers, balls, droplets, and other objects when quadratic drag dominates. It does not include buoyancy, wind, tumbling, changing area, compressibility, or altitude-dependent density. Very small particles at low Reynolds number may follow linear rather than quadratic drag. Long high-altitude falls need numerical integration because gravity and density vary along the path. Treat published drag coefficients as estimates and match them to the expected orientation. For engineering or safety decisions, validate inputs experimentally and use a more complete trajectory model when environmental or shape changes are important.