Analytic Solutions for Drag and Magnus forces

When a spinning object moves in air, it is affected by three forces: the gravitational force, the drag force, and the Magnus force. The equations of motion for such an object are nonlinear, making it difficult to find analytic solutions. Hence, numerical solutions using computers are often preferred. This paper provides two examples of analytic solutions of the equations of motion with the drag and Magnus forces: nearly vertical motion and zero-gravity motion. In these two cases, we find analytic solutions for the velocity and the position of the object, avoiding mathematical difficulties. It is suitable for university students studying physics.