How do you code air resistance?

How do you code air resistance?

Add one term to the acceleration: gravity pulls down at $g$, and drag pushes back with $k v^2$, where $k = \tfrac{1}{2}C_d\rho A/m$. Step the speed forward with $a = g - kv^2$ and the object speeds up, then levels off at the terminal velocity $\sqrt{g/k}$ — exactly what happens in a real fall.

This is the code companion to How free fall works, which derives the drag force and terminal velocity.

The whole simulation

const g = 9.81, rho = 1.225;   // m/s², kg/m³
const m = 80, CdA = 0.7;        // skydiver: mass (kg), drag coefficient × area (m²)

const k = (0.5 * rho * CdA) / m;      // drag acceleration per v²
const vTerminal = Math.sqrt(g / k);   // 42.8 m/s

// s.v = downward speed (m/s), s.y = distance fallen (m)
function step(s, dt) {
  const a = g - k * s.v * s.v;   // gravity minus quadratic drag
  s.v += a * dt;                 // update velocity first…
  s.y += s.v * dt;               // …then position with the new velocity
}

// Closed-form solution for quadratic drag, used as the test oracle
const exact = (t) => ({
  v: vTerminal * Math.tanh((g * t) / vTerminal),
  y: ((vTerminal * vTerminal) / g) * Math.log(Math.cosh((g * t) / vTerminal)),
});

const s = { v: 0, y: 0 };
for (let t = 0; t < 30; t += 0.01) step(s, 0.01);
console.log(s, exact(30));

Does it match the exact solution?

For this skydiver the terminal velocity is $42.8$ m/s, reached to within 5% after $8.0$ s and within 1% after $11.5$ s. Compared with a drag-free fall ($98.1$ m/s after $10$ s), drag has already held the speed to $41.9$ m/s. Distance fallen in $30$ s:

Going to two dimensions with wind

In 2D, drag opposes the motion relative to the air. If the air moves with velocity $\vec w$, use $\vec v_{rel} = \vec v - \vec w$ and apply $\vec a_{drag} = -k\,|\vec v_{rel}|\,\vec v_{rel}$ to both components. A tailwind then reduces drag on the ball; a headwind increases it. The Ball Gravity simulation keeps things simpler: wind is just a constant horizontal acceleration, with no drag.

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