Rotational deflection of a falling body

In an Earth-fixed rotating reference frame, the horizontal displacement of a freely falling body receives contributions from Coriolis acceleration and centrifugal acceleration. A short-fall approximation substitutes the unperturbed downward velocity into the Coriolis term and the Earth's surface position into the centrifugal term. If gravitational acceleration excludes centrifugal acceleration, both contributions must be included explicitly.

A body released from rest relative to Earth at the equator has unperturbed downward velocity $v_U=-gt$. The Coriolis acceleration in local Earth coordinates gives eastward acceleration $a_E=-2{\Omega }_E{v}_U=2{\Omega }_{E}gt$. Integration gives $x_E={\Omega }_{E}g{t}^3/3$ and impact time $t_f=\sqrt{2h/g}$, proving the title formula. This is first order in Earth's rotation, neglecting air resistance and the height dependence of the effective local gravitational acceleration. For a short fall at latitude $\phi$, its leading eastward term is multiplied by $\cos \phi$.