Jet impingement flow is known to generate one of the highest single-phase heat transfer rates, with potential for micro-electronics cooling applications. Although free-surface jets have been studied extensively, existing models are either too complex for practical use or do not consider all relevant parameters, such as the impinging jet's velocity profile. Recently the authors have shown that the stagnation zone heat transfer is dictated by the jet's centerline velocity upon impingement, and that going between the limiting cases (uniform vs. parabolic profiles, laminar flow) corresponds to a two-fold increase in heat transfer. In the present study, which is motivated by cooling at microscales (predominantly laminar flows), this simplified analysis is extended leading to a first-order analytical approximation, which is valid not only for the limiting cases but over the entire profile range. Thereby, the development of the jet flow both in the nozzle (pipe-Type) and subsequently during its flight (before impingement) is incorporated in this model over a broad range of parameters. For validation of the model, as well as for additional insight into the governing physics, direct numerical simulations were conducted. Through which it is shown that the jet's velocity profile and its evolution during free "flight" are dependent on the level of the flow's upstream development in the nozzle, both of which depend on a single self-similar scale: distance travelled normalized by the nozzle diameter and Reynolds number. This one-way coupling requires incorporation of both stages of development for an accurate description, as done in the present model. During pipe-flow, the first-order model employs a more-rapid development rate than during jet-flight (due to the additional pressure-driven flow) - converging to more complex, well-known models, within a few pipe diameters (for Re=200 to 2300). During flight, the model describes velocity profile relaxation, which is dominated by viscous diffusion and assisted by jet contraction. Jet contraction is dependent on the emerging velocity profile and liquid-vapor surface tension. For most relevant conditions surface tension is negligible, under which the first-order model centerline velocity decay prediction agrees well with both present simulations and previous works. Thereby, the present work lays the foundation for a simpler, more useable model for predicting heat transfer under an impinging free-surface jet, over a wide range of conditions (various liquids, pipe-Type nozzles of different lengths, flowrates and nozzle-To-plate distances), as part of an ongoing study into micro-jet array heat transfer.