Analytical Investigation of Projectile Motion in Midair

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. All the basic functional dependencies of the problem are described by elementary functions. There is no need for 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.


INTRODUCTION
The problem of the motion of a projectile thrown at an angle to the horizon in midair has a long history. It is one of the great classical problems. The number of works devoted to this task is immense. It is a constituent of many introductory courses of physics. With zero air drag force, the analytic solution is well known. The trajectory of the projectile is a parabola. In real tasks, such as throwing a ball, the impact of the medium is taken into account. Usually quadratic drag low is used. In this case the mathematical complexity of the task strongly grows. The problem probably does not have an exact analytic solution. Therefore the attempts are being continued to construct approximate analytical solutions for this problem. In the given paper an analytic approach is used for the investigation of the projectile motion in a medium with quadratic resistance. The proposed analytical solution differs from other solutions by simplicity of formulae, ease of use and high accuracy. The proposed formulas make it possible to study the motion of a projectile in a medium with the resistance in the way it is done for the case without drag. These formulae are available even for first-year undergraduates.
The problem of the motion of a projectile in midair arouses interest of authors as before [1][2][3][4][5][6][7][8]. For the construction of the analytical solutions various methods are usedboth the traditional approaches [1], 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 a simple approximate analytical formulae under the quadratic air resistance is of great methodological and educational importance. The purpose of the present work is to give a simple formulas for the construction of the trajectory of the projectile motion with quadratic air resistance. 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 (Fig.1). Air resistance force is proportional to the square of the velocity of the projectile and is directed opposite the velocity vector. For the convenience of further calculations, the drag force will be written as 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 wacceleration vector of the projectile. Differential equations of the motion, commonly used in ballistics, are as follows [9] 2 gkV θ sin g 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, Here V0 and θ0 are the initial values of the velocity and of the slope of the trajectory respectively, t0 is the initial value of the time, x0, y0 are the initial values of the coordinates of the projectile (usually accepted The derivation of the formulae (2) is shown in the well-known monograph [10]. The integrals on the right-hand sides of formulas (3) cannot be expressed in terms of elementary functions. Hence, to determine the variables t, x and y we must either integrate system (1) numerically or evaluate the definite integrals (3).

OBTAINING AN ANALYTICAL SOLUTION OF THE PROBLEM
The task analysis shows, that equations (3)  θ θ π f θ tg θ . Therefore, it can be assumed that a successful approximation of this function will make it possible to calculate analytically the definite integrals (3) with the required accuracy. In Ref. [1], the function () From the conditions (4) we find  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: For the variable t we get: Here we introduce the notation: Thus, the dependence   t  has the following form: Consequently, the basic functional dependencies of the problem       Using formulas (5) -(7), we find: Then formulas (5) -(7) can be rewritten as: We note that formulas (5) -(7) also define the dependences in a parametric way.

THE RESULTS OF THE CALCULATIONS. FIELD OF APPLICATION OF THE OBTAINED SOLUTIONS
Proposed formulae have a wide region of application. We introduce the notation       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. 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 formulae make it possible to study projectile motion with quadratic drag force even for first-year undergraduates.