From snowballs to ICBM’s

The elliptic range RE of a ballistic missile is the length of arc between the two intercepts of the circle of the earth with the elliptic orbit of the projectile. When expanded in powers of the projection velocity v, RE is given by RE=2v² sinϑ cosϑ/e+O (v⁶), where e is the eccentricity of the orbit. Units are such that GM=1 and the radius of the Earth is one. The lead term is thus the correct form for the solution to the elementary ballistic problem. It is shown that this lead term not only describes the flight of a snowball, but can also give a good account of the range of an ICBM. Thus, for a projection velocity of v=√2/2 (half the escape velocity) the lead term gives the range for any angle of projection with less than a 2% error.