An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method

We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object ( quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 x 10(3) similar to 2 x 10(5) ( for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution.