General Solution of Two-dimensional Projectile Motion

In this study, two-dimensional projectile motion is considered under the 4 effect of a general power law model of air resistance. Classically, a 5 projectile is treated as a point mass with mass m moving in a uniform 6 gravitational field. The projectile is launched from the ground with an angle 7 α to horizon. the drag force is assumed to be proportional to the speed 8 raised to the power n. The analysis of the problem is performed using 9 Cartesian coordinates. A general exact parametrical solution (with respect 10 to the angle of motion) is derived for any power n, following simple steps: 11 1) find the speed in the direction of the axis x (horizontal – no gravity); 2) 12 find the vertical component of the speed; 3) find the time; 4) find the 13 horizontal position of the projectile; and finally, 5) find the vertical position 14 of the projectile. Steps 1) and 2) give explicit closed form equations and the 15 rest are given by exact integrals which can be solved numerically. In this 16 study spreadsheet calculation are performed using trapezoidal rule of 17 integration. The cases of motion in a vacuum and linear drag law are used 18 to check the accuracy of the numerical calculations. The importance of the 19 proposed study is three-fold: a) The method of the derived solution is new, 20 and couldn’t be found elsewhere; b) The derived equations make it possible 21 to use spreadsheets for presenting the subject (no programming capabilities 22 is required), and thus, serve as a tool to enhance teaching; c) The derived


Introduction
to the projectile's speed, there exists an analytic solution [7,8]. For the general case where the drag coefficient is proportional to any power of the speed, there 1 is no closed form solution. The quadratic drag case was studied extensively 2 using different approaches including numerical calculations; analytic 3 approximations; simulations, and by introducing exact integrals. [5 -7, 9 -13]. 4 The projectile's motion with a general power law of air resistance was studied 5 by using path coordinates (a projectile's speed and angle of motion) [13]. The 6 projectile's position was presented parametrically using exact integrals. 7 In this study, the two-dimensional projectile's motion with a general power 8 law of air resistance model is reconsidered by using cartesian coordinates. The  Newton's second law for the projectile's motion is written as follows: Where ⃗ is the velocity vector and is given by: And the projectile's speed is given by: After dividing by the mass m, equation (1) is rewritten for x components 17 of the acceleration as follows: Where is the proportionality constant D divided by the mass m and is given Similarly, the y component of equation (1) for y component of the 1 acceleration, and is given by: For convenience, nondimensional variables are used. The subscript s 6 denotes dimensional scales as specified in table1.
Following the definitions that are given in table 1, equation (4) is rewritten 11 in dimensionless form and is given by: 14 15 Similarly, equation (6) is rewritten in dimensionless form and is given by: In order to simplify the solution method, equation (7) is divided by * and 20 rewritten as follows: The same simplification is used as before such that equation (8) is divided 24 by * and rewritten as follows: By subtracting equation (10) from equation (9) and after proper 1 mathematical manipulation, the following equation is derived: Notice that the ratio between the speed components is given by: Where θ is the angle of motion. By using the chain rule of differentiation (11) is rewritten respectively and is given by: Similarly, by using the chain rule of differentiation ( integrating the velocity components with respect to time. 37 The results derived in this section are used as building blocks for the 38 general model of two-dimensional projectile motion.

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A general model of a projectile's motion in two dimensions 1 2 In this section the procedure to find the position of the projectile versus 3 time is outlined in subsection 3.1.

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General procedure 6 7 1) Solve equation (15) for u. The solution can be found by the method of 8 separation of variables. The differential equation for u is given by: The general solution for u as a function of θ is given by: 2) Solve equation (16) for time by the method of separation of variables. The 13 non-dimensional time is given by: Similarly, the vertical speed component is calculated from equation (8). The 2 non-dimensional vertical speed is given by: Then, the non-dimensional vertical position is derived by substituting 7 equation (24) in equation (21), and completing the integration, thus, the y* is 8 given by: Calculations based on the equations of the general model 13 By substituting * = 0 in equation (18), the non-dimensional horizontal 14 speed is shown to be a constant and is given by: Then, by means of equation (12), the vertical speed is given by: The parametric relation between the nondimensional time and the angle θ 23 is derived by means of equation (19) and is given by: By eliminating tan ( ), equation (28) is rewritten in the form:  In this case, the horizontal acceleration after substituting n = 1 in equation 3 (7), is given by: Similarly, the vertical acceleration after substituting n = 1 in equation (8), 8 is given by: By integrating equation (30) and using the appropriate initial condition 13 (entry 8 in table 1), the horizontal speed is given by: Similarly, by integrating equation (31) and using the appropriate initial 18 condition (entry 9 in table 1), the vertical speed is given by: Now, starting at the origin, the horizontal position of the projectile is 23 derived by integrating equation (32) and is given by: By repeating the same procedure, the vertical position of the projectile is 28 derived by integrating equation (33) and is given by: Calculations based on the equations of the general model 33 By applying the procedure of the general model for the case n = 1, the 34 horizontal speed is found from equation (18) and is given by (step 1): Based on step 2, the non-dimensional time is calculated from the 39 following integral (after substituting the expression for u): By completing the integration, the non-dimensional time is given by: In fact, after substituting equation (36) in equation (38), the expression for 5 the horizontal speed (see equation (32)) is retrieved. 6 The vertical speed component is obtained by dividing equation (33)  shown to be given by equation (33).

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To complete the calculations, the projectile's position is found exactly as For cases of ≥ 2 the general procedure to calculate the parameters of the 31 projectile's motion is followed as was described previously (steps 1-4). The The vertical component of the velocity vector is calculated by using 1 equation (12), and the numerical integration is performed by means of the 2 trapezoidal rule method.     37 38 For the ideal case, projectile's motion in vacuum or without air resistance,   It is important to note that for the motion in a vacuum, the horizontal 11 speed is a constant, thus the error in finding the x coordinate is the same as the 12 error in calculating the time. This is depicted in figure 3.

Relative error% between analytic non-dimensional vertical position and its numric calculation
It is important to note that calculating the speed components were based on 1 analytic formulas equations (22) and (27). The projectile's position depends on the non-dimensional air resistance 21 coefficient. Figure 6 shows the non-dimensional horizontal position as a 22 function non-dimensional time. Similarly, Figure 7 shows the non-dimensional vertical position as a 4 function non-dimensional time. important to note that this plot has a parabolic shape for a projectile's motion in 2 a vacuum only, as was discovered by Galileo. As was pointed out early, the accuracy depends on the step size. In  Based on this estimation method, the relative error in calculating the position 10 coordinates is given in table 3.  coordinate is inversely related to friction with air (see figure 9). Finally, the vertical coordinate is plotted vs. the horizontal coordinate to n 9 = 2 (see figure 11). The general procedure could be used as a tool for education. For example, 1 one might ask: "What is the time elapsed for reaching maximum height? "  The projectile's motion with a general power law model of air resistance 3 was studied using Cartesian coordinates. The equations of motion were derived 4 from Newton's second law. The acceleration equations in the horizontal and 5 vertical directions are non-linear and coupled ordinary differential equations. 6 The solution of these equations is simplified by a proper choice of the 7 independent variables. In fact, decoupling the equations of motion was 8 achieved by choosing the horizontal velocity and the angle of motion as 9 independent variables. It is shown that the velocity components were derived 10 analytically (see step 1 in the general procedure). The time (see step 2), the 11 horizontal coordinate (see step 3) and the vertical coordinate (see step 4) were 12 obtained by solving exact integrals. Trapezoidal rule quadrature was used to 13 estimate the integrals.
14 Three cases of motion were considered: motion in a vacuum; motion 15 under air resistance with n = 1; and motion under air resistance with n = 2.