Through a simple, classical, energy conservation analysis, we propose a finite distance reinterpretation of the standard energy fraction definition used for the single differential cross section (SDCS) for the electron-hydrogen $S$ wave ionization process. The energy modification is due to the fact that, at finite distances from the nucleus, the continuum electrons have to overcome the remaining potential energy to be completely free. As a consequence, the flux formula for extracting - at finite distances - SDCS is also modified. Different from the usual observations, the proposed corrections yield finite and well behaved SDCS values also at the asymmetrical situation where one of the continuum electrons carries all the energy while the other has zero energy. Results of calculations performed at various impact energies, for both singlet and triplet symmetry, are presented and compared favorably with benchmark theoretical data. Although we do not know how, we believe that finite distance effects should strongly affect the evaluation of the flux and consequently the SDCS, also in the full electron-hydrogen case.