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. Differently 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.