For almost a century, velocity dependent scattering problems are solved in the context of Einstein's Special Relativity theory. Most interesting problems involve non-uniform motion, which is heuristically justified by assuming the validity of the "instantaneous velocity" approximation. The present study attempts to provide a consistent postulational foundation by introducing boundary conditions based on the Lorentz force formulas. The methodology used here is dubbed "reverse engineering": Being aware of the relativistic results, we show that they are replicated, (at least) to the first order in β = v/c by the present method. Specific problems are discussed to demonstrate the power of the method, and pave the way to future research in this problem area. Specifically, by realizing that at the boundary we deal with signals, which are derived from waves, only the latter being subject to the wave equations, it is feasible to apply boundary conditions and construct appropriately the scattered waves in space. It is shown that the present approach is also consistent with the Minkowski constitutive relations which are exploited for solving problems where the medium moves parallel with respect to the boundaries.
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