Precision calculation of the projectile trajectory of a spherical object considering air resistance

To study the precise trajectory of a sphere under conditions where air resistance is considered, differential methods were employed for real-time and accurate theoretical calculations of motion trajectories. The focus was on the effects of the sphere’s mass, diameter, initial velocity, and projection angle on its trajectory, which were compared with scenarios without air resistance. Finally, equations relating these affecting parameters to the horizontal distance or the time of flight until impact were constructed. The results indicate that the mass effect on the impact of air resistance is the most significant, followed by the initial velocity, then the diameter, and lastly the projection angle. A larger mass, a lower initial velocity, and smaller diameter can mitigate the influence of air resistance. When projected at an upward angle, there exists an optimal projection angle that maximizes the horizontal distance of impact. In cases of downward projections within the angle range [75°, 90°), situations were found where the horizontal distance of impact was greater when air resistance was considered.