![]() ![]() The path of the projectile is called the trajectory and can be modeled by a quadratic equation, assuming the only force acting on the motion is gravity (no friction). A projectile is any moving object that is thrown near the Earth's surface. Li, Trajectory instability and convergence of the curvilinear motion of a hard projectile in deep penetration. ![]() Scholte, The Influence of the Magnus Effect in Tennis, Bachelor’s Project Applied Mathematics (University of Groningen, Groningen, 2017) Turkyilmazoglu, Highly accurate analytic formulae for projectile motion subjected to quadratic drag. Sarafian, Impact of the drag force and the magnus effect on the trajectory of a baseball. Volni, Uniform projectile motion: dynamics, symmetries and conservation laws. Robinson, The motion of an arbitrarily rotating spherical projectile and its application to ball games. Seifert, A review of the Magnus effect in aeronautics. Ebaid, Analysis of projectile motion in view of fractional calculus. Chudinov, Approximate analytical investigation of projectile motion in a medium with quadratic drag force Int. Eberhard, On the ascent and descent times of a projectile in a resistant medium. Konosevich, An estimate of the error of the linearized equations of motion of an axisymmetric projectile. Jiang, Computational fluid dynamics study of magnus force on an axis-symmetric, disk-type AUV with symmetric propulsion. Oshima, Negative magnus effect on a rotating sphere at around the critical Reynolds number. Plakhov, Magnus effect and dynamics of a spinning disc in a rarefied medium. Everett, Release angle for attaining maximum distance in the soccer throw-in. Vogel, Comparative Biomechanics: Life’s Physical World (Princeton University Press, Princeton, 2003) Lions, The backwards jump of the box mite. Almaguer, On the aerodynamic forces on a baseball, with applications. Vlasak, Magnus and Drag Forces Acting on Golf Ball, Colloquium Fluid Dynamics (Institute of Thermomechanics AS CR, Prague, 2007), pp. Kommer, The Magnus effect and the American football. Subic, Review of tennis ball aerodynamics. Pallis, in Sports Ball Aerodynamics: Effects of Velocity, Spin and Surface Roughness Materials and Science in Sports, ed. Coppinger, Electrically-launched mm-sized hypervelocity projectiles. Xiao, Numerical investigations on the oblique water entry of high-speed projectiles. ![]() Gao, Modeling and calculation method of target damage based on multi-attitude flying projectile in space intersection. Gupta, Deformation and ballistic performance of conical aluminum projectiles impacting thin aluminum targets: influence of apex angle. Wang, Influence of yawing force frequency on angular motion and ballistic characteristics of a dual-spin projectile. Vaziri, An efficient method for continuous measurement of projectile motion in ballistic impact experiments. Examples are eventually provided and discussed for the accuracy and reliability of the given formulae against the full numerical simulation.ĭ. The presented formulae can be employed to estimate and optimize the characteristic kinematics of the projectile. Otherwise, full solution is further representable under the restriction of a uniform Magnus force with a still quadratic drag force. In the case of simultaneous existence of quadratic drag and Magnus forces together, the governing motion equation is highly nonlinear, permitting only to the analytical perturbation solutions. An expression for the vertical distance of the projectile thrown from a fixed position is also accounted for measuring the impacts of Magnus effects on the maximum height, the striking velocity to the ground and angle of stroke of the projectile during motion. Closed-form solutions for the speed of the object are provided either when the quadratic drag force is negligible or when the quadratic Magnus force is negligible. In this paper, it is targeted to present exact and approximate solutions to the motion of fired projectile in air subject to the Magnus effect. ![]()
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