Terminal Velocity (2D): force balance, v(t) and distance graphs

Presets

Parameters

Mass Cross-section area Air density Drag coefficient Cd Time speed

Live readouts

TimeSpeedTerminal speedWeight m·gDragNet accel.FallenTime const. τ=vt/g

Model check

Press "Run check".

The science

Quadratic drag

Newton's second law along the fall: m dv/dt = m·g − ½·Cd·ρ·A·v². Setting dv/dt = 0 gives v_t = √(2mg / (Cd·ρ·A)).

The exact solution from rest is v(t) = v_t·tanh(g·t / v_t) and the distance fallen is y(t) = (v_t²/g)·ln cosh(g·t / v_t). After about 3 time constants (τ = v_t/g) the speed is within 1 % of v_t.

What this view adds

The 3D page shows the body falling; here you see the force balance as arrows of true relative length, the v(t) curve against the exact tanh curve, and the drag curve approaching the weight line.

Simplifications

Constant air density and Cd (real air thins with altitude and Cd depends on Reynolds number), a point mass, vertical motion only, no wind or buoyancy. Cd values are rough skydiving figures, not measurements. Integration is classical RK4 with a fixed step of 5 ms.