Falling-body simulator with quadratic air drag. A body released from rest accelerates under gravity (g = 9.81 m/s²) while drag, F = ½ρCdAv², grows with the square of speed until it exactly balances gravity at the terminal velocity v_term = sqrt(2mg / (ρ C_d A)). The equation of motion has a closed-form solution v(t) = v_term · tanh(g t / v_term), so no numerical integration is needed. Compare a feather, a basketball-like ball and a belly-to-earth skydiver side by side to see how the mass-to-drag ratio controls how quickly each one settles into free fall at constant speed.
This simulation drops an object from rest and follows its speed as gravity and quadratic air drag fight for control. Newton's second law for a falling body reads m dv/dt = mg − ½ρCdAv², where the drag term grows with the square of speed while gravity's pull stays constant. Early on, drag is tiny and the object accelerates at almost exactly g = 9.81 m/s². As speed builds, drag catches up until the two forces exactly balance and acceleration drops to zero — the object has reached terminal velocity, v_term = √(2mg / (ρCdA)). Because this particular differential equation is separable, it has an exact closed-form solution rather than needing step-by-step numerical integration: v(t) = v_term·tanh(gt/v_term), and the distance fallen is y(t) = (v_term²/g)·ln(cosh(gt/v_term)). The hyperbolic tangent starts at zero, rises almost linearly like free fall, then flattens smoothly into the terminal-velocity plateau.
What decides how fast an object "settles" is the ratio of mass to drag area, mCdA — heavier, denser, more streamlined objects need a much higher speed before drag can balance their weight, so they take longer (in time) to approach their terminal velocity even though, relative to that velocity, all objects follow the same tanh curve. A feather with almost no mass reaches its very low terminal velocity in a fraction of a second; a skydiver falling belly-first needs several seconds and roughly 250–300 metres of fall to get close to their much higher terminal velocity of around 50–55 m/s. Use the preset buttons or sliders to compare a feather, a basketball-sized ball and a skydiver side by side, and switch on Compare Mode to overlay all three v(t) curves and falling columns at once. Note that this is a deliberately different topic from the site's parachute simulation: the parachute sim focuses on the transient mechanics of chute deployment — the sudden jump in drag area and deceleration spike when a canopy opens — while this page is about the general drag-vs-gravity race for an unchanging object shape, comparing how differently sized and shaped bodies approach their own steady falling speed.
What does this simulation actually show?
It shows one or more objects falling from rest under gravity and quadratic air drag, both as an animated falling column and as a velocity-vs-time graph. As the object speeds up, drag grows until it balances gravity, and the velocity curve flattens onto a dashed terminal-velocity line. Telemetry reports the live speed, terminal velocity, instantaneous acceleration, distance fallen and the time needed to reach 95% of terminal velocity.
What is terminal velocity and how is it calculated?
Terminal velocity is the constant speed a falling object eventually reaches once drag exactly cancels gravity. Setting mg equal to the drag force ½ρCdAv² and solving for v gives v_term = √(2mg / (ρCdA)). It depends on the object's mass, frontal area, drag coefficient and the density of the air (or other fluid) it is falling through.
Why does velocity follow a tanh curve instead of a straight line?
The equation of motion m dv/dt = mg − ½ρCdAv² is a separable first-order ODE whose exact solution is v(t) = v_term·tanh(gt/v_term). Near t = 0 the hyperbolic tangent is almost linear, matching free fall under gravity alone; as t grows, tanh saturates toward 1, so v smoothly approaches v_term and never overshoots it.
Setting tanh(gt95/v_term) = 0.95 and solving for t95 gives t95 = (v_term/g)·artanh(0.95), and artanh(0.95) = ½·ln(1.95/0.05) ≈ 1.8318. So t95 ≈ 1.8318·v_term/g. A higher terminal velocity means a proportionally longer time (and much greater fall distance) is needed to get close to it.
Instantaneous acceleration is a(t) = g·(1 − (v(t)/v_term)²). At t = 0, v = 0 so a = g, the full acceleration of gravity. As v approaches v_term, the ratio v/v_term approaches 1 and a smoothly approaches zero — the net force on the object vanishes exactly at terminal velocity, which is why the velocity curve flattens rather than crossing the dashed asymptote.
Terminal velocity scales with √(m/(CdA)). A feather has extremely low mass but a relatively large frontal area for that mass, so its terminal velocity is tiny — often under 1–2 m/s. A skydiver has orders of magnitude more mass per unit of frontal area, so gravity needs a much higher speed before drag can catch up, giving a terminal velocity around 50–55 m/s in a belly-to-earth position.
The mass, frontal area, drag coefficient and air-density sliders let you override any preset and immediately recompute v_term and the tanh curve. Compare Mode runs the feather, ball and skydiver presets simultaneously, drawing one falling column per object and overlaying their three v(t) curves and terminal-velocity asymptotes on the same graph so you can directly see how differently each one approaches its own steady speed.
The parachute simulation focuses on the deployment event itself — the sudden increase in drag area when a canopy opens and the resulting deceleration spike as a jumper transitions from a fast free fall to a slow, parachute-assisted descent. This simulation instead isolates the general physics of drag approaching terminal velocity for a single, unchanging object shape, and is built around comparing how mass, area and drag coefficient change the shape and timescale of that approach across very different falling objects.
The closed-form tanh solution is mathematically exact for constant mass, constant drag coefficient, constant frontal area, and constant air density, which is an excellent approximation for objects falling through a few hundred metres of roughly uniform low-altitude air. It omits secondary effects such as changes in air density and gravity with altitude, orientation changes (tumbling), wind, and compressibility effects that matter only at much higher speeds.