This study delves into the influence of muon positioning on particle behavior within atoms, exploring scenarios where the muon resides inside, outside, or partially inside the nucleus of a hydrogen atom. The observed behavior of the differential cross section (DCS) in these contexts yields significant insights into atomic interactions. When the muon is within the nucleus, a marked decrease in the DCS is noted as the electron's momentum undergoes changes. This suggests a substantial impact of the muon on scattering behavior, akin to phenomena in laser-assisted electron-hole scattering and classical scattering. Conversely, when the muon is outside the nucleus, a sharp decrease in DCS is observed at low momentum changes, in contrast to the behavior observed when the muon is within the nucleus. In cases where the muon is partially internal, an intermediary pattern emerges: exponential decay at lower energies and a gradual decline beyond 1.5 MeV. This behavior bridges the scenarios depicted with the muon inside and outside the nucleus. The findings underscore the pivotal role of muon placement in shaping scattering dynamics within atoms. This enhanced understanding of atomic interactions carries profound implications for the advancement of nuclear physics and our comprehension of the fundamental constituents of matter. The development and computational analysis of corresponding equations were facilitated through the MATLAB student package, bolstering the robustness of the study's conclusions.