Simple and convenient analytical formulas for studying the projectile motion in midair

Abstract

Here is studied a classic problem of the motion of a projectile thrown at an angle to the horizon. The air drag force is taken into account as the quadratic resistance law. An analytic approach is used for the investigation. Equations of the projectile motion are solved analytically. The basic functional dependencies of the problem are described by elementary functions. There is no need to study the problem numerically. The found analytical solutions are highly accurate over a wide range of parameters The motion of a baseball and a badminton shuttlecock are presented as examples.

Analytical construction of the projectile motion trajectory in midair

“…Based on this idea, in [7,8] their high-precision analytical solutions for the variables x, y, and t were obtained over a sufficiently large interval of variation of the variable θ. In the present work, these solutions are continued up to the limit value θ = −π/2.…”

“…Under the condition, θ < 0, nother approximation is required because the function f (θ) is odd. In [7,8] two approximations of the function f (θ) on the whole interval −π/2 < θ < π/2 were proposed. The first approximation uses a second-order polynomial, the second approximation uses a third-order polynomial.…”

“…Following other authors, we call this approach the analytic approach. From the point of view of the applied methods of solution, the present research is a development of the approaches used in [7,8].…”

Analytical Investigation of Projectile Motion in Midair

“…In [3], a very fruitful idea was proposed for calculating integrals (3). Based on this idea, in [4,5] there were obtained high-precision analytical solutions for the variables x, y and t over a sufficiently large interval of variation of the variable θ.…”

“…Following other authors, we call this approach the analytic approach. From the point of view of the applied methods of solution, the present research is a development of the approaches used in [4,5]. The conditions of applicability of the quadratic resistance law are deemed to be fulfilled, i.e.…”