Simulate free fall with drag using numerical integration. Calculate terminal velocity, real vs ideal fall distance, time-velocity tables, and compare drag profiles for different objects.
Real-world free fall differs dramatically from the idealized vacuum case because of air resistance (drag). A skydiver in belly-down position reaches terminal velocity of about 55 m/s (200 km/h) after roughly 12 seconds, while a feather reaches its terminal velocity of only 0.5 m/s almost instantly.
The drag force on a falling object is F_d = ½ρv²C_dA, where ρ is air density, v is velocity, C_d is the drag coefficient, and A is the cross-sectional area. As the object accelerates, drag increases with v² until it equals the gravitational force mg. At this point — terminal velocity — the object stops accelerating and falls at constant speed.
This calculator uses numerical integration to simulate the fall second by second, accounting for changing velocity and optionally varying air density with altitude. It provides comparison tables for different objects and drag profiles. Check the example with realistic values before reporting. The simulation keeps drag, terminal velocity, and altitude context together so you can compare a real fall against the ideal vacuum case more directly.
Free fall with drag requires solving a nonlinear differential equation that has no simple closed-form solution for the distance-time relationship. This calculator uses numerical simulation to produce accurate time-velocity-distance profiles, compare real vs ideal trajectories, and instantly compute terminal velocities for any object. Keep these notes focused on your operational context.
Drag Force: F_d = ½ρv²C_dA Net acceleration: a = g − F_d/m = g − (ρv²C_dA)/(2m) Terminal Velocity: v_t = √(2mg / (ρC_dA)) ISA Air Density: ρ(h) = 1.225 × (T(h)/288.15)^4.256 where T(h) = 288.15 − 0.0065h Numerical integration (Euler): v(t+dt) = v(t) + a·dt, d(t+dt) = d(t) + v·dt
Result: Terminal velocity = 47.8 m/s (172 km/h), distance = 614 m in 15 s
An 80 kg skydiver (belly-down, Cd=1.0, A=0.7 m²) at sea level: v_t = √(2×80×9.81/(1.225×1.0×0.7)) = 47.8 m/s. After 15 s with drag, falls 614 m vs 1,103 m without drag — a 44% reduction.
Terminal velocity represents the fundamental limit that air resistance places on falling speed. The balance between gravity (constant) and drag (increasing with v²) means that every falling object eventually reaches a maximum speed. For a human skydiver, this takes about 12 seconds and approximately 450 meters of altitude.
The equation v_t = √(2mg/(ρCdA)) reveals which factors matter: increasing mass or decreasing drag coefficient or area raises terminal velocity. This is why streamlined shapes fall faster and why heavier skydivers fall faster than lighter ones at the same body position.
This calculator uses the International Standard Atmosphere (ISA) model for air density, which decreases exponentially with altitude. During a high-altitude jump, the falling speed initially exceeds the sea-level terminal velocity because the thin air produces less drag. As the jumper descends into denser air, they decelerate even while aerodynamic forces increase.
Understanding drag-limited free fall is essential in many fields: ballistics (bullet trajectory), aerospace (re-entry heating), meteorology (raindrop size distribution), and industrial processes (spray drying, particle separation). The same physics governs sediment settling in water, where the fluid density is much higher.
Terminal velocity is the constant speed reached when the drag force equals gravitational force (mg). At this point, net acceleration is zero and the object falls at constant velocity. It depends on mass, shape, and air density.
The differential equation ma = mg − ½ρv²CdA has an analytical solution for velocity vs time (hyperbolic tangent), but distance vs time requires numerical methods. The Euler method steps through time in small increments for accurate results.
Air density decreases with altitude (about 1.225 kg/m³ at sea level, 0.41 kg/m³ at 10 km). Lower density means less drag and higher terminal velocity — this is why Felix Baumgartner exceeded Mach 1 falling from 39 km.
By changing body position, a skydiver changes both Cd and A. Belly-down (Cd≈1.0, A≈0.7 m²) gives ~55 m/s. Head-down (Cd≈0.4, A≈0.15 m²) gives ~90 m/s. The ratio is roughly √(CdA) between positions.
With dt = 0.01 s, the Euler method gives accuracy within 0.1% for typical free fall scenarios. More sophisticated methods (Runge-Kutta) are rarely needed for 1D falling problems.
For dense objects like humans, buoyancy in air is negligible (less than 0.1% of weight). For very light objects like balloons, buoyancy matters significantly but this calculator focuses on drag-dominated falling.