Obtaining the Analytical Solution of the External Ballistics Problem with the Resistance Force Approximations

The ballistic problem of body motion in a resistance medium in the homogeneous gravity field is considered. For the resistance force, a quadratic dependence on speed is used. Assumptions are made about the low of the trajectory, the limited range of the movement speed, and the position that follows from them about the constancy of the drag coefficient. Additionally, the approximation of the sine and cosine functions of the trajectory angle is applied. The adoption of these assumptions made it possible to obtain a solution to the equations system of body motion in a compact analytical form. The formulas obtained for the velocity components and coordinates were tested using the numerical solution of exact differential equations without applying the described assumptions. Testing showed that, within the framework of the assumptions, the agreement between the results of the analytical approximate and conditionally exact numerical solutions is very good. The final formulas derived for velocities and coordinates can be applied when performing practical calculations for applied problems of various subjects.