# New constant-velocity component paths for objects moving in vertical space

## DOI:

https://doi.org/10.15282/jmes.18.3.2024.7.0804## Keywords:

Dynamics of Motion, Thrust forces, Path equation, Ordinary differential equations, Perturbation iteration methods, Numerical solutions## Abstract

Paths of vehicles under restricted conditions are of technological interest in navigation engineering. One such restriction may be to fix one of the velocity components during motion. For objects moving in a two-dimensional vertical space, the differential equations determining the paths of the objects for which one of the velocity components remains constant are derived. First, the no thrust force case is investigated. The two paths in which the velocity components remain constant are determined by finding exact solutions of the associated differential equations. While the constant *x*-component case produces the well-known parabolic solution, the constant y-component case reveals a new solution called the 2/3 rule. Then, the differential equations for an object moving with constant *x* and *y* velocity components are derived separately for the constant-magnitude thrust force case. Since the equations inherit high nonlinearities, exact analytical solutions cannot be obtained for the constant-magnitude thrust force case. Instead, approximate solutions obtained by the Perturbation Iteration Method are compared with the Runge-Kutta numerical solutions. Within the range of validity, the approximate solutions can be employed to determine the path instead of the numerical solutions. The approximate analytical solutions would reduce the computational cost of integrating the original numerical solutions. The study may find applications in determining the paths of flying objects such as projectiles, rockets, and aerial vehicles.

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*J. Mech. Eng. Sci.*, vol. 18, no. 3, pp. 10181–10191, Sep. 2024.

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