= Ballistic orbit insertion with constant gravity
{title2=$h_b=h_o-v_o^2\tan^2\theta/(2g)$}
In the flat-surface, constant-<gravitational acceleration> model, a coasting projectile conserves horizontal <velocity>. If it arrives at the top of its trajectory with horizontal <speed> $v_o$, its earlier <velocity> at <angle> $\theta$ above the horizontal has vertical component $v_o\tan\theta$. Its remaining height gain is therefore $v_o^2\tan^2\theta/(2g)$. The displayed burnout height follows. This is a local <projectile motion> approximation, not an exact <Kepler orbit>. Computing $v_o=\sqrt{g(R_E+h_o)}$ assumes constant gravitational magnitude also at the orbit; computing $v_o=\sqrt{g_0R_E^2/(R_E+h_o)}$ instead assumes an <inverse-square law> with surface value $g_0$. Those assumptions must be distinguished.
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