Projectile Motion and Free Fall Calculator with Air Drag

The translational equations are derived here from Newton's second law. With horizontal wind $w$, the relative velocity is $\mathbf u=(v_x-w,v_y)$, and the drag force is $\mathbf F_D=-\tfrac12\rho C_D A|\mathbf u|\mathbf u$. Thus $\dot x=v_x$, $\dot y=v_y$, $\dot v_x=-b(v_x-w)$ and $\dot v_y=-g-bv_y$, where $b=\rho C_D A|\mathbf u|/(2m)$. This is the NASA Glenn drag equation applied opposite the air-relative velocity. The sphere frontal area is $\pi d^2/4$.

For the optional smooth-sphere model, $Re=\rho d|\mathbf u|/\mu$. Schiller–Naumann uses $C_D=24(1+0.15Re^{0.687})/Re$ below $Re=1000$ and $C_D=0.44$ above it (OpenFOAM 12 implementation). The low-speed force is evaluated in its nonsingular viscous form, including at zero speed.

The NASA Glenn metric troposphere fit uses $T_f=288.14-0.00649z$, $p=101290(T_f/288.08)^{5.256}$ and $\rho=p/(286.9T_f)$ in SI units, with $z=z_{ground}+y$. These rounded fit constants use NASA's 273.1 offset convention; this is not an exact implementation of the 1976 standard atmosphere. Sutherland viscosity is $\mu=1.458\times10^{-6}T^{3/2}/(T+110.4)$ Pa·s (Cambridge).

Dormand–Prince integration uses relative accuracy $10^{-6}$, with a time scaling based on the initial drag rate to resolve rapid deceleration. Cubic Hermite interpolation locates ground contact and the apex between integration nodes; the apex is inserted before plotting. A 5° sweep brackets the best angle, followed by parabolic refinement at 0.5° spacing; boundary maxima use a bounded numerical search. Each coarse candidate records whether it reaches the ground; the refined trajectory is checked again for altitude, Mach number, sphere Reynolds number, and touchdown. Treat the displayed angle as approximately 0.1° resolution. The vacuum path is calculated analytically.

Still-air terminal speed at the landing ground solves $b(0,v_t)v_t=g$; it reduces to $v_t=\sqrt{2mg/(\rho C_D A)}$ for a constant coefficient. It remains a reference with drag switched off. Animation reads the cached path; it does not rerun the integration or angle search.

Before a drag-based angle search, conservative bounds certify its entire 0–90° domain. The height bound is $H=h_0+v_0^2/(2g)$. Let $U_0$ be the greatest initial air speed in that domain, and let $V_{t,max}$ use the least density and a lower bound on the drag coefficient. Then $U\le\min(\sqrt{U_0^2+2gh_0},\max(U_0,V_{t,max}))$. This bounds Mach number using the lowest temperature and bounds Reynolds number using ground density and the lowest viscosity. Temperature has no upper-altitude clamp; an out-of-domain atmosphere is never accepted for a recommendation.

For touchdown certification, let $B$ bound the drag rate $b$ over this domain. Descending speed satisfies $q'\ge g-Bq$, so descent distance after time $t$ is at least $gt^2/[2(1+Bt)]$. Hence the fall-time bound is $(HB+\sqrt{(HB)^2+2gH})/g$; adding $v_0/g$ bounds the entire flight. If any bound exceeds the model limits, no angle recommendation is offered. These conservative checks can withhold a search even when some individual launch angles remain valid.

• Two-dimensional point-mass motion over a flat horizontal landing surface; positive height is upward.
• Constant gravity and a steady horizontal wind. Wind has no effect when drag is disabled.
• Fixed mass, frontal area, and posture throughout the flight. No lift, Magnus force, rotation, or deployment events.
• Dry-air altitude fit, or a user-specified constant density and temperature. Ground altitude is distinct from release height.
• With drag enabled, buoyancy is neglected and ball mass must exceed 20 times the displaced air mass, limiting omitted buoyancy to below 5% of weight. With drag disabled, this restriction does not apply to the vacuum trajectory; the retained terminal-speed reference has a separate buoyancy warning.
• Default ball dimensions and coefficient are editable illustrative inputs, not a certified sports-ball model. Skydiver defaults follow OpenStax Example 6.17.

Use for subsonic preliminary estimates, with the whole altitude-based flight below 11,000 m. The smooth-sphere correlation is restricted conservatively to $Re<2\times10^5$; its drag-crisis, roughness, and spin effects are not modelled. Constant coefficients must be appropriate to the chosen body and flow; they are not universally valid.

With drag enabled, the numerical solver supports local still-air terminal speeds of at least 1 m/s and ground contact within 600 s. A failed condition explains which inputs to change. Air-relative Mach numbers above 0.3 trigger a compressibility warning. The launch-angle search covers 0–90° and maximizes signed +x displacement; drop modes have no best launch angle. A drag-based angle search is offered only when conservative bounds certify all angles in its 0–90° domain. Otherwise the selected flight remains available if valid, the recommendation says unavailable, and the validated search curve is omitted. This does not mean every individual angle is invalid.

All chart axes use the SI units printed on them; the metric/US switch converts the physical input and result-card units. Body icons are enlarged, but the plotted trajectories use one geometric scale. The fixed terminal-speed line refers to the landing-ground atmosphere, not the changing local equilibrium throughout a high-altitude fall.

Not a parachute, human-impact, or jump-planning tool. It does not predict landing safety.

Vacuum trajectories use the analytical equations before any numerical drag-solver restriction. Mass, diameter, frontal area, buoyancy, air-relative Mach number, and the atmosphere's altitude limit do not constrain the ideal vacuum path. Any Reynolds-number, compressibility, or buoyancy limitations of the retained terminal-speed reference are reported separately.