Mechanical · Experiment

Projectile with Air Drag

In a vacuum every projectile traces a perfect parabola. Add air resistance and the path shrinks, tilts, and comes down steeper than it went up — real life, not the textbook.

with air dragvacuum (ideal parabola)velocitydrag force

Controls

Range (drag)— m
Range (vacuum)— m
Max height (drag)— m
Flight time (drag)— s
Terminal velocity— m/s
Impact speed · angle—
—
About this experiment ›

What you are looking at

The same launch, shown two ways: the pale dashed curve is the ideal vacuum path (a perfect parabola), and the bright curve is the real path with air resistance. The moving dot follows the real trajectory, with its velocity (green) and the drag force (orange) drawn on it. Compare where the two land.

The drag force

As an object pushes through air, the air pushes back. For anything moving quickly the resistance grows with the square of the speed and always points opposite to the motion:
F_drag = −k · v · |v|  (k = ½ ρ C_d A)
Here k bundles up the air density ρ, the object's drag coefficient C_d, and its cross-sectional area A. So the full motion is gravity plus drag:
m a = −m g ĵ − k v |v|
There is no tidy formula for this path — it has to be computed step by step (this uses a 4th-order Runge–Kutta integrator). Set k = 0 and the bright curve snaps exactly onto the vacuum parabola.

What air does to the flight

Drag steals energy the whole way, so compared with the vacuum path the projectile flies a shorter range, reaches a lower peak, and — the striking part — comes down more steeply than it went up. The trajectory is no longer symmetric: on the way up drag and gravity both fight the motion, but on the way down drag opposes gravity, so the fall is slower and steeper. Because of this, the range-maximising launch angle in air is a bit below 45°, not exactly 45°.

Terminal velocity

In a long fall, drag eventually balances gravity and the speed stops increasing — the terminal velocity:
k v_t² = m g  →  v_t = √(m g / k)
A dense, streamlined object (cannonball) has a huge terminal velocity and barely notices the air; a light, bluff object (beach ball) has a tiny one and is dominated by drag. Try the presets to feel the difference.

Things to try

Start from the vacuum case (k = 0), then raise the air resistance and watch the bright curve peel away from the dashed parabola and steepen on descent. Compare a heavy cannonball with a light beach ball at the same launch. Then hunt for the angle that gives the greatest range with drag on — you'll find it sitting a little under 45°.