Extension of application field of analytical formulas for the computation of projectile motion in midair

Extensão do campo de aplicações de fórmulas analíticas para os cálculos do movimento de um projétil no ar

ARTIGOS GERAIS

Extension of application field of analytical formulas for the computation of projectile motion in midair

Extensão do campo de aplicações de fórmulas analíticas para os cálculos do movimento de um projétil no ar

Peter Chudinov¹

Perm State Agricultural AcademYI Perm, Russian Federation

ABSTRACT

The classic problem of the motion of a point mass (projectile) thrown at an angle to the horizon is reviewed. The air drag force is taken into account with the drag factor assumed to be constant. An analytical approach is used for the investigation. Application field of the previously obtained approximate analytical formulas has been expanded both in the upward launch angle and in the direction of increase of the initial speed of the projectile. The motion of a baseball is presented as an example. It is shown that in a sufficiently wide ranges of initial velocity and launch angle the relative error in calculating the distance of the ball does not exceed 1%.

Keywords: spherical object, quadratic drag force, analytical formula, relative error.

RESUMO

O problema clássico do movimento de um ponto material (projétil) jogado com determinado ângulo em relação à direção horizontal é revisto nesse artigo. A força de resistência do ar é levada em conta, supondo que o coeficiente de arrastamento seja constante. Uma abordagem analítica é utilizada para essa investigação. O campo de aplicação das fórmulas analíticas aproximadas obtidas anteriormente foi expandido tanto em relação ao ângulo de lançamento quanto em relação aos valores da velocidade inicial do projétil. O movimento de uma bola de basebol é apresentado como exemplo. Mostra-se que, para intervalos suficientemente amplos de velocidade inicial e ângulo de lançamento, o erro no cálculo da distância percorrida pela bola não é superior a 1%.

Palavras-chave: objeto esférico, força quadrática de arrastamento, fórmulas analíticas, erro relativo.

1. Introduction

The problem of the motion of a point mass (projectile) thrown at an angle to the horizon has a long hi&tory. The number of works devoted to this task is immense. It is a constituent of many introductory courses of physics. This task arouses interest of authors as before [1-6]. Wilh zero air drag force, lhe analylic solution is well known. The trajectory of the projectile is a parabola. ln situations of practical interest, such as throwing a ball, taking into account the impact of the medium the quadratic resistance law is usually used. ln that case the problem probably does not have an exact analytic solution and therefore in most scientific publicalions ii is solved numerically [7-11]. Analylic approaches to the solution of the problem are not sufficiently advanced. Meanwhile, analytical solutions are very convenient for a straightforward adaptation to applied problems and are especially useful for a qualitative analysis. Therefore the attempts are being continued to construct analytical solutions (even approximate) for this problem. Comparativly simple approximate analytical formulas to study the motion of the point mass in a medium with a quadratic drag force have been oblained using such an approach [12-14]. These formulas make it possible to carry out a complete qualitative and quantitative analysis without using numerical integration of differential equations of point mass motion. The proposed analytical solution differs from other solutions by easy formulas, ease of use and acceptable accuracy. It is intended to study the motion of a baseball, and other similar objects.

Present article is largely initiated by the interesting work of Hackborn [4]. ln Ihis paper lhe accuracy of various analytical approximations of the projectile trajectories was investigated in the calculation of their distance in a fairly wide ranges of variation of the initial velocity and launch angle. The purpose of present paper is to extend the application field of the formulas [12-14] and lo compare lhe accuracy of Ihese formulas for calculating the projectile range with the resulls oblained in Ref. [4]. ln lhe paper under consideration the term "point mass" means the center of mass of a smooth spherical object of finite radius r and cross-sectional area S = πr². The conditions of applicability of the quadratic resistance law are deemed to be fulfilled, i.e. Reynolds number Re lies within 1 x 10³< Re < 3 x 10⁵ [5].

2. Equations of point mass motion and analytical formulas for basic parameters

Suppose that the force of gravity affects the point mass together with the force of air resistance R (Fig. 1), which is proportional to the square of the velocity of the point and directed opposite the velocity vector. For the convenience of further calculations, the drag force wiU be written as R = mgkV² . 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 point mass has the form

mw= mg+R,

where w -acceleration vector of the point mass. Differential equations of the motion, a commonly used in ballistics, are as follows [15]

Here V is the velocity of the point mass, θ is the angle between the tangent to the trajectory of the point mass and the horizontal, x, y are the Cartesian coordinates of the point mass, k = = const is the proportionality factor, ρa is the air density, C_dis the drag factor for a sphere, and S is the cross-section area of the object (Fig. 1). The first two equations of the system (1) represent the projections of the vector equation of motion for the tangent and principal normal to the trajectory, the other two are kinematic relations connecting the projections of the velo city vector point mass on the axis x, y with derivatives of the coordinates.

The well-known solution [15] of Eqs. (1) consists of an explicit analytical dependence of the velocity on the slope angle of the trajectory and three quadratures

Here V₀ and θ₀ are the initial values of the velocity and 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 point mass (usua11y accepted t₀ = X₀ = Y₀ = 0), The derivation of the formulas (2) is shown in the we11-known monograph [16],

The integrals on the right-hand sides of Eqs. (3) cannot be expressed in terms of elementary functions. Hence, to determine the variables t, x and y we must either integrate Eqs. (1) numerically or evaluate the definite integrals (3). To avoid these procedures, comparatively simple approximate analytical formulas for the eight basic parameters of point mass motion were obtained in [12-14] (Fig. 1). The four parameters correspond to the top of the trajectory, four - to the point of drop. We will give a complete summary of the formulas for the maximum height of ascent of the point mass H, motion time T , the velocity at the trajectory apex V_a, flight range L, the time of ascent t_a, the abscissa of the trajectory apex x_a, impact angle with respect to the horizontal θ₁ and the final velocity V₁

In formulas (4) V₀ and θ0 are the initial values of the velocity and the slope of the trajectory of the point mass, respectively. Formulas (4) enable us to calculate the basic parameters of motion of a point mass directly from the initial data V₀, θ0, as in the theory of parabolic motion. With zero drag (k =0), formulas (4) go over into the respective formulas of point mass parabolic motion theory.

As an example of the use of formulas (4) we calculated the motion of a baseball with the following initial conditions

V₀ = 45 m/s; θ0 = 40º; k = 0.000548 s²/m², g = 9.81 m/s².

The results of calculations are recorded in Table 1. The second column shows the values of parameters obtained by numerical integration of the motion Eqs. (1) with the fourth-order Runge-Kutta method. The third column contains the values calculated by formulas (4). The deviations from the exact values of parameters are shown in the fourth column of the table.

3. Extension of application field of the formulas (4)

For baseball typical values of the drag force coeficient k is about 0.0005/0.0006 s²/m²[4, 9]. We introduce the notation p = kV₀². The dimensionless parameter p has the following physical meaning - it is the ratio of air resistance to the weight of the projectile at the beginning of the movement. Formulas (4) have a bounded region of application. The main characteristics of the motion H, T, V_a, L, t_a, x_a, θ1, V₁ have accurate to within 2-3% for values of the launch angle, initial velocity and the parameter p from ranges

We transform the formulas (4) so as to improve the accuracy of calculating the basic measure of the motion -range L in the entire range of launch angles and at values of the initial velocity and the parameter p larger than compared with the ranges of Eq. (5)

For this we consider the structure of the range formula L = V_aT. According to this formula, range of the motion is defined as the product of the velocity at the top of the trajectory V_aon the time motion T Therefore, to incre ase the accuracy of computation of the range L it is necessary to incre ase the accuracy of calculating the parameters V_aand T. Let's start with the incre ase the accuracy of the parameter T. ln turn, according to the second of the formulas (4), the time of the projectile motion T is determined by parameter H. The formula for the maximum height of ascent of the projectile H is the most important of all the formulas (4). When the launch angle θ0 increases, height H computed according to the first formula (4) is behind the exact value of this parameter. The exact value of H can be obtained by integrating the equations of motion (1). The greatest noncoincidence occurs at an angle of throwing 80 = 90º. It is known [15] that the maximum height attained by the point mass at throwing with the initial conditions Vo, θ0 = 90º, is given by

Formula (4) for H at θ0 = 90º gives the value of

We introduce the notation

The quantity Δh -is a mismatch between the exact and approximate values of the height at θ0 = 90º. We form the function f (p), equal to the ratio (8) to (7)

The graph of f(p) is shown in Fig. 2.

The graph shows that when the force of air resistance incre ases (parameter p grows), formula (4) for H loses accuracy when θ₀= 90º. The sarne holds for large angles of throwing θ₀> 70º. To eliminate this drawback, we modify the formula (4) for H. Let it be like this

Here ƒ₁ (θ₀) is a function of the launch angle satisfying ƒ₁ (0º) = 0, ƒ₁ (90º) = 1. The choice of this function is quite arbitrary. Let the function ƒ1 (θ₀) be given by the following empirical formula

Graph of this function is shown in Fig. 3. Such a structure of formula (10) allows more accurate calculation of the height H at high angles of throwing θ > 70º. When θ₀= 90º, formula (10) gives the exact value of the height (7).

Now we transform the second factor V_ain the range formula. The parameter V_ais the velocity of the projectile at the top of the trajectory calculated by the formulas (2) at θ= 0º. Instead of the values of the velocity at the top of the trajectory at θ= 0º we calculate the velocity of the projectile at some close point of the trajectory defined by the angle of inclination θ_a. We define angle θ_a , measured in degrees, as a function of the parameters p and θ₀: θ_a = ƒ₂ (p,θ₀). The choice of this function is arbitrary. However, the value of θ_a should be positive in order to incre ase the velocity of the projectile compared with the value of it at the top of the trajectory. This follows from the well-known fact that on the upward trajectory velocity is greater than at the topo We define the function ƒ₂ as follows

Now under V_awe understand the value determined by the relation

Thus, the formulas (4) for H and V_aare replaced by the formulas (10), (12). ln the absence of resistance of the medium (k = O) formulas (10), (12) tum into the corresponding formulas of the parabolic motion theory. ln addition, formulas (4) make it possible to obtain simple approximate analitical expressions for the basic functional relationships of the problem y(x), y(t), y(θ), x(t), x(θ), t(θ) [13]. For example, the trajectory equation y( x) has the form

This formula shows that for the construction of dependence y(x) we need to know three parameters: H, L and x_a, which are determined by formulas (4). This dependence y(x) provides a shift of apex of the path to the right and has a vertical asymptote, as it is in case of the of projectile trajectories in the air. ln the absence of the resistance L = 2x_aand formula (13) go into the corresponding formula of the theory of parabolic motion.

4. The results of calculations and their companson

The formulas (10), (12), (13) were used to calculate the motion of a baseball. It is convenient to calculate the coefficient of resistance k in these formulas by means of the terminal velocity of a ball (Fig. 1): k = 1/V²_term. We used the following typical values of k (for V_term= 40 m/s) and the acceleration of gravity g

The initial velocity, launch angle and parameter p varied in the ranges (6). When calculating the range of the ball, the relative error was computed as a function of initial velocity and launch angle. Distance calculated according to formulas (4), (10), (12), was compared with the exact value ofthis parameter. The exact values of range were obtained by integration of the equations ofmotion (1) by the fourth-order Runge-Kutta method. The calculation results are shown in Fig 4. This figure is the main result of this article.

The horizontal plane in Fig 4 corresponds to the zero value of the relative error. Figure 4 shows that in the covered scope (6) the relative error is less than 1%. Figure 5 shows the projectile trajectory y(x) for values of V_o= 80 m/s, θ0 = 60º. The solid line is obtained by integrating Eqs. (1), the broken line is constructed by formulas (4), (10), (12), (13). It is obvious that these formulas approximate a precise trajectory quite well. ln paper [4] well-known analytical approximations of the trajectory of Parker [7], Littlewood [17], Lamb [18] are quoted, and also Hackborn gave his original formula, close to above-mentioned ones. These approximations were used in [4] to calculate the distance of baseball motion in the ranges of initial velocity and launch angle 10 < V_o< 80 m/s, 15º < θ₀< 75º for values of Eq. (14). The relative error in calculating the distance for these approximations was calculated in Ref. [4]. It was found by Hackborn that in these ranges the relative error reaches 4.5% ÷ 6.5%. These values are much higher than the accuracy of 1% obtained with the help of formulas (4), (10), (12), (13).

5. Conclusions

The proposed modification of the formulas [12-14] allows us to significantly expand the range of their use in the studying of the projectile motion in midair. All the basic parameters of the motion and functional dependencies are still described by simple analytical relations. ln addition the numerical values of the required quantities are determined with high accuracy. Thus, the formulas (4), (10), (12), (13) make it possible to study the motion of a point mass in a medium with resistance in the way it is done for the case of no drag.

Recebido em 12/4/2012

Aceito em 23/6/2012

Publicado em 18/2/2013

• [1] G.F.L. Ferreira, Revista Brasileira de Ensino de Física 23,271 (2001).
• [2] A.D.S. Bruno and J.M.O. Matos, Revista Brasileira de Ensino de Física 24,30 (2002).
• [3] E.N. Miranda, S. Nikolskaya and R. Riba, Revista Brasileira de Ensino de Física 26,125 (2004).
• [4] W. Hackborn, Canadian Applied Mathematics Quarterly 14,285, (2006)
• [5] A.Vial, European Journal of Physics 28,657 (2007).
• [6] J. Benacka, International Journal of Mathematical Education in Science and Technology 41,373 (2010).
• [7] G. Parker, American Journal ofPhysics 45,606 (1977).
• [8] H. Erlichson, American Journal of Physics 51,357 (1983).
• [9] A. Tan, C. Frick and O. Castillo, American Journal of Physics 55,37 (1987).
• [10] C. Groetsch, American Journal of Physics 65,797 (1997).
• [11] J. Lindemuth, American Journal of Physics 39,757 (1971).
• [12] P. Chudinov, International Journal of Mathematical Education in Science and Technology 41,92 (2010).
• [13] P. Chudinov, International Journal of Sports Science and Engineering 5,27 (2011).
• [14] P. Chudinov, European Journal of Physics 25,73 (2004).
• [15] B. Okunev, Ballistics (Voyenizdat, Moscow, 1943).
• [16] S. Timoshenko and D. Young, Advanced Dynamics (McGraw-Hill Book Company, New York, 1948).
• [17] J. Littlewood, Mathematical Spectrum 4,80, (1971).
• [18] H. Lamb, Dynamics (Cambridge University Press, London, 1923).

Publication Dates

• Publication in this collection
  07 May 2013
• Date of issue
  Mar 2013

History

• Received
  12 Apr 2012
• Accepted
  18 Feb 2013
• Reviewed
  23 June 2012