Dot = projectile. Colour = Mach (blue subsonic, amber transonic, red supersonic). Dashed grey = vacuum parabola.

Long-Range Projectile Lab - 2D

Side-view trajectory coloured by Mach, drag curve, drift and elevation sweep.

-Range (m)

-Apex (m)

-Flight time (s)

-Impact speed (m/s)

-Max Mach / impact Mach

-Lateral drift (m)

-Vacuum range (m)

-Range lost to drag (%)

Elevation sweep

Which elevation gives the longest range, and how far is it from the 45 degrees of the vacuum parabola? Wind, Coriolis and spin are switched off for the sweep.

Independent checks

The science

Point mass, RK4 (dt = 4 ms). Same module as the 3D page.

a = g + a_drag + a_Coriolis, a_drag = -(1/2) rho Cd(M) A/m |v_rel| v_rel, v_rel = v - v_wind, M = |v_rel|/c, c = sqrt(gamma R T), a_Coriolis = -2 Omega x v.

Vacuum check. Without drag the range is R = v cos(th) (v sin(th) + sqrt(v^2 sin^2(th) + 2 g h)) / g and the best angle is atan(v / sqrt(v^2 + 2 g h)), which is 45 degrees at launch height 0. The first check compares the RK4 result with this closed form.

Drag curve. Cd(M) is a generic smooth textbook curve of a streamlined round, flat below Mach 0.8, rising steeply through Mach 1 and peaking near 1.1. It is not a table for any real object. Watch the moving dot on the Cd chart: the sharp loss of speed happens when the path crosses the transonic bump.

Wind, spin, Coriolis. Wind enters through the relative velocity (drift is about wind x (t - x/v0)). Spin drift uses the empirical rule 1.25 (Sg + 1.2) t^1.83 inches, calibrated only for flights of a few seconds and scaled with air density. Coriolis is integrated in the rotating frame (Moon: 27 times weaker than Earth; Mars: similar to Earth).

Simplifications. No lift, Magnus force, yaw, wind shear or Earth curvature; flat ground at launch height. This is a physics teaching model of a generic test body, not a tool for any real equipment.