We have studied a mesoscopic ring threaded by a magnetic flux that increases linearly with time. The ring is partially coherent, such that conduction electrons in the ring will encounter incoherent scatterings. In addition, the electrons encounter elastic scatterings due to the presence of an impurity in the ring. We have adopted a S-matrix model, as proposed by Buttiker, for the incoherent scatterings in this time-dependent situation. This allows us to treat the incoherent scatterings, the elastic scatterings and the coherent inelastic processes on the same footing. We have solved the problem exactly. Our results demonstrate that, in the case of a weak impurity, the lower the energies of the electrons that emanate out of incoherent scatterings, the greater will be their net contribution to the dc component Idc of the induced current. In the case of a strong impurity, however, Idc alternates between regions of zero and nonzero values as the chemical potential μ increases. The peak value of Idc in the nonzero region increases with μ. We find that these regions of zero, and nonzero, Idc correspond closely with the gaps, and the bands, respectively, of a one-dimensional energy band. All these characteristics arise from the fact that the electrons traversing the ring have their energies shifted gradually until their energies fall upon a forbidden region, where they suffer total reflection. This total reflection at the forbidden region does not occur in a ring that has a constant flux. Rather, it results from the nonadiabatic effect of the changing flux. The evolution of the nonadiabatic effects in the intermediate impurity regime has also been investigated.