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.