Highly precise analytic solutions for classic problem of projectile motion in the air

Here is studied a classic problem of the motion of a projectile thrown at an angle to the horizon. The number of publications on this problem is very large. 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. All 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 is presented as examples.


Introduction
The problem of the motion of a projectile in midair arouses interest of authors as before [1][2][3][4][5][6][7][8]. The number of publications on this problem is very large. Together with the investigation of the problem by numerical methods, attempts are still being made to obtain the analytical solutions. Many such solutions of a particular type are obtained. They are valid for limited values of the physical parameters of the problem (for the linear law of the medium resistance at low speeds, for short travel times, for low, high and split angle trajectory regimes and others). For the construction of the analytical solutions various methods are usedboth the traditional approaches [1,3,4], and the modern methods [2,5]. All proposed approximate analytical solutions are rather complicated and inconvenient for educational purposes. In addition, many approximate solutions use special functions, for example, the Lambert W function. This is why the description of the projectile motion by means of simple approximate analytical formulas under the quadratic air resistance is of great methodological and educational importance.
The purpose of the present work is to give simple formulas for the construction of the trajectory of the projectile motion with quadratic air resistance. In this paper, two variants of approximation of the sought functions (the projectile coordinates) is realized. It allows to construct a trajectory of the projectile with the help of elementary functions without using numerical schemes. Following other authors, we call this approach the analytic approach. The conditions of applicability of the quadratic resistance law are deemed to be fulfilled, i.e. Reynolds number Re lies within 1×10 3 < Re < 2×10 5 .

Equations of projectile motion
We now state the formulation of the problem and the equations of the motion according to [8].
Suppose that the force of gravity affects the projectile together with the force of air resistance R (see figure 1). Air resistance force is proportional to the square of the velocity V of the projectile and is directed opposite the velocity vector. For the convenience of further calculations, the drag force will

R mgkV 
. Here m is the mass of the projectile, g is the acceleration due to gravity, k is the proportionality factor. Vector equation of the motion of the projectile has the form where w -acceleration vector of the projectile. Differential equations of the motion, commonly used in ballistics, are as follows [9] 2 sin dV g gkV dt Here V is the velocity of the projectile, θ is the angle between the tangent to the trajectory of the projectile and the horizontal, x, y are the Cartesian coordinates of the projectile, In the present paper we propose two approximations of the function The first approximation uses a second-order polynomial, the second approximation uses a third-order polynomial. Approximation of the function The function f3 () is formed by two odd functions, and therefore it is applicable over the whole interval

Analytical solutions for the approximation f 2 (θ)
Now the quadratures (3)  For the coordinate x we obtain: in case of θ ≤ 0.
Here we introduce the following notation: x  has the following form: We integrate the second of the integrals ( Here we introduce the following notation: Thus, the dependence   y  has the following form: Consequently, the basic functional dependencies of the problem       Then formulas (5) - (7) can be rewritten as: The angle of incidence of the projectile 1  is determined from the condition   We note that formulas (5) -(7) also define the dependences in a parametric way.

Analytical solutions for the approximation f 3 (θ)
Now let the function ( ) f q in formula (2) We integrate the second of the integrals (3). For the coordinate y we obtain: Here we introduce the following notation: Thus, the dependence   y  has the following form: Consequently, the basic functional dependencies of the problem     , xy  are written in terms of elementary functions. The third integral (3) cannot be taken in elementary functions. However, estimates for the parameters T and ta can be made using the formulas of [6]. The angle of incidence of the projectile 1  is determined from the condition   1 0 y   . Using formulas (10) -(11), we find: We note that formulas (10) -(11) also define the dependence   y y x  in a parametric way.

The results of the calculations. Field of application of the obtained solutions
Proposed formulas have a wide region of application. We introduce the notation   In this case the values of the parameter p vary from 1 to 9.  identical trajectories are described with various analytical formulas (5) -(6) and (10) -(11).
As an example of a specific calculation using formulas (5) - (7), we give the trajectory and the values of the basic parameters of the motion L, H, T, xa , ta , θ1 for shuttlecock in badminton. Of all the trajectories of sport projectiles, the trajectory of the shuttlecock has the greatest asymmetry. This is explained by the relatively large value of the drag coefficient k and, accordingly, by the large values of the parameter p. Initial conditions of calculation are k = 0.022 s 2 /m 2 ; V0 = 50 m/s ; θ0 = 40° ; p = 55. Table 1. Basic parameters of the shuttlecock movement. The second column of Table 1 contains range values calculated analytically with formulae (8) - (9). The third column of Table 1 contains range values from the integration of the equations of system (1). The fourth column presents the error of the calculation of the parameter in the percentage. The error does not exceed 2 %. Thus, a successful approximation of the function   f  made it possible to calculate the integrals (3) in elementary functions and to obtain a highly accurate analytical solution of the problem of the motion of the projectile in the air.

Conclusion
The proposed approach based on the use of analytic formulae makes it possible to simplify significantly a qualitative analysis of the motion of a projectile with the air drag taken into account. All basic variables of the motion are described by analytical formulae containing elementary functions. Moreover, numerical values of the sought variables are determined with high accuracy. It can be implemented even on a standard calculator. Thus, proposed formulas make it possible to study projectile motion with quadratic drag force even for first-year undergraduates.