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The movement of a body subjected to a drag force proportional to the square of the velocity is of great importance in the field of mechanical engineering, since it is a realistic model in the regime of high velocity of the flow. In this paper, the trajectory of a projectile in quasi-vertical launch with quadratic drag is analyzed using the technique of matched asymptotic expansions. Through the asymptotic analysis, approximate analytical expressions for the projectile movement are found. The asymptotic solutions are compared with exact numerical results. [ABSTRACT FROM AUTHOR]

Rapidly increasing demand for electronic devices and electric vehicles necessitates efficient and high power and energy density storage devices. In this context, we synthesized and optimized reduced Graphene Oxide (rGO) nanosheets intercalated with Fe3O4 nanoparticles using an in situ method for supercapacitor and Lithium-ion battery performance. The synthesized materials were well characterized using FESEM, XRD, TGA, and RAMAN spectroscopy. The optimized 25-wt% Fe3O4@rGO nanocomposite in the form of asymmetric supercapacitor exhibits a high specific capacitance of 310 F/g at 10 A/g and evaluated an energy density of 39.375 Wh/Kg and power density of 1041.6 W/Kg, respectively. The same Fe3O4@rGO material, in the form of symmetric capacitor, shows the 19 F/g at a current density 10 A/g using a 3-M KOH as electrolytic solution with an energy density of 0.95 KWh/Kg at power density of 403 W/Kg for planar supercapacitor devices. Furthermore, the optimized Fe3O4@rGO, evaluated for lithium-ion battery (LIB) anode performance, shows a high specific capacity of 290 mAh/g at 0.5 A/g and 212 mAh/g at 1 A/g current densities while maintaining stability over 500 cycles. The incorporation of Fe3O4 in rGO nanosheets results in the enhancement of the storage performance synergistically. The influence of Fe3O4 on the surfaces of rGO as interfaces contributes to faradaic capacitive reaction and rGO for non-faradaic electric double-layer capacitive through electronic/ionic transport for the electrochemical behavior of the nanocomposite. Overall, the highly efficient and low-cost Fe3O4@rGO nanocomposite show superior capacitance and LIB anode performances are the key findings which can be potential electrodes in supercapacitor-battery hybrid storage devices for both high energy density and high power density. [ABSTRACT FROM AUTHOR]