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. 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. 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. 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. Otherwise, full solution is further representable under the restriction of a uniform Magnus force with a still quadratic drag force. The presented formulae can be employed to estimate and optimize the characteristic kinematics of the projectile. Examples are eventually provided and discussed for the accuracy and reliability of the given formulae against the full numerical simulation.

中文翻译：

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包含马格努斯效应的空中弹丸运动的精确解和近似解

在本文中，其目标是提出受马格努斯效应影响的空中射击弹丸运动的精确和近似解。当二次拖动力可以忽略不计时或当二次马格努斯力可以忽略不计时，提供物体速度的闭式解。从固定位置投掷的弹丸垂直距离的表达式也可以用来衡量马格努斯效应对运动过程中弹丸最大高度，对地面的打击速度以及弹丸的冲程角度的影响。在同时存在二次阻力和马格努斯力的情况下，控制运动方程是高度非线性的，仅允许解析扰动解。除此以外，在均匀的马格努斯力和仍然是二次拖曳力的限制下，完全解可以进一步表示。提出的公式可用于估计和优化弹丸的运动学特性。最终提供并讨论了针对完整数值模拟的给定公式的准确性和可靠性的示例。