Projectile Motion with Air Drag

What you are seeing: three projectiles fired simultaneously at the same speed $v_0$ and angle $\theta$ , subject to different drag laws. The yellow projectile is in vacuum (no drag); the cyan one experiences Stokes drag $F = -b\mathbf{v}$ ; the orange one experiences quadratic drag $F = -c|\mathbf{v}|\mathbf{v}$ . All three leave the launcher at the same time but land at different ranges.

Without drag the trajectory is a parabola with range $R = v_0^2 \sin(2\theta) / g$ (max at $\theta = 45^\circ$ ). Stokes drag is linear and falls off more gently with speed; quadratic drag grows faster at high speed, so it bites harder near launch. Slide the speed and angle to see the asymmetry: at high speeds quadratic drag dominates; at low speeds Stokes drag wins.

Figure 1. Projectile with three drag laws (vacuum, Stokes, quadratic). Method: RK4 integration; the capture spans the vacuum flight.

v_0 (m/s)20

angle45 deg

drag1.0x

speed2

WHAT TO TRY

Watch the three balls land: the vacuum shot flies farthest, the drag shots fall short and come down steeper than they went up.
In the lower plot, the vacuum range peaks exactly at 45 degrees. With drag the best angle slides below 45, marked by the dots on each curve.
Push the launch speed up: quadratic drag bites harder and the gap to the vacuum shot widens.
Sweep the drag slider from 0 to 2.5x: at zero all three curves collapse onto the vacuum parabola; turn it up and the drag optima slide further below 45 degrees while the ranges shrink.