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Revista Brasileira de Ensino de Física

Print version ISSN 1806-1117On-line version ISSN 1806-9126

Rev. Bras. Ensino Fís. vol.40 no.1 São Paulo  2018  Epub Aug 14, 2017

https://doi.org/10.1590/1806-9126-rbef-2017-0145 

Articles

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

Peter Chudinov1  * 

Vladimir Eltyshev1 

Yuri Barykin1 

1Engineering Faculty, Perm State Agricultural Academy, Perm, Russian Federation


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.

Keywords: classic problem; projectile motion; quadratic drag force; analytical formulas

1. Introduction

The problem of the motion of a projectile in midair arouses interest of authors as before [1-8]. 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 used - both 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 formulas 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. In this paper, one of the 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 .

2. Equations of projectile motion

We now state the formulation of the problem and the equations of the motion according to [7]. 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 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 projectile has the form

Figure 1 Basic motion parameters. 

mw=mg+R,

where w - acceleration vector of the projectile. Differential equations of the motion, commonly used in ballistics, are as follows [9]

dVdt=-gsinθ-gkV2,dθdt=-gcosθV,dxdt=Vcosθ,dydt=Vsinθ. (1)

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,

k=ρacdS2mg=1Vterm2=const,

ρa is the air density, cd is the drag factor for a sphere, S is the cross-section area of the object, and Vterm is the terminal velocity The first two equations of the system (1) represent the projections of the vector equation of motion on the tangent and principal normal to the trajectory, the other two are kinematic relations connecting the projections of the velocity vector projectile on the axis x , y with derivatives of the coordinates.

The well-known solution of system (1) consists of an explicit analytical dependence of the velocity on the slope angle of the trajectory and three quadratures

Vθ=V0cosθ0cosθ1+kV02cos2θ0fθ0-fθ,fθ=sinθcos2θ+lntanθ2+π4, (2)
x=x0-1gθ0θV2dθ,y=y0-1gθ0θV2tanθdθ,t=t0-1gθ0θVcosθdθ. (3)

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 t0=x0=y0=0). The derivation of the formulas (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).

3. Obtaining an analytical solution of the problem

The analysis of the task shows, that equations (3) are not exactly integrable owing to the complicated nature of function (2). The odd function fθ is defined in the interval -π2 < θ < π2 . 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.

The Ref. [1] presents a simple approximation in the mathematical sense of a function f(θ) by a second-order polynomial of the following form (polynomial is with respect to a function tanθ)

faθ=a1tanθ+b1tan2θ.

An analysis of the problem shows that it is convenient to approximate the function f(θ) only by polynomials of the second or third degree. The first-order polynomial does not provide the required accuracy of the approximation. Polynomials of higher orders do not allow us to calculate the integrals (3) in elementary functions. The polynomial of the second order approximates the function f(θ) well only on a bounded interval 0,θ0. Under the condition θ <0, another approximation is required because the function f(θ) is odd. Therefore, the question of using a second-order polynomial for a given problem requires a separate study. As already noted, the function faθ well approximates the function fθ only on the limited interval 0,θ0 , since the function faθ contains an even term. Therefore, in the present paper we approximate the function fθ on the whole interval -π2 < θ < π2 with a function f1θ of the following form

f1θ=a1tanθ+b1tan3θ.

The function f1(θ) is formed by two odd functions. The coefficients a1 and b1 can be chosen in such a way as to smoothly connect the functions fθ and f1θ to each other with the help of conditions

f1θ0=fθ0,f1θ0=fθ0. (4)

From the conditions (4) we find

a1=12cosθ0+3lntan(θ02+π4)2tanθ0,b1=12tan2θ01cosθ0-lntan(θ02+π4)tanθ0.

Such a function f1θ well approximates the function fθ throughout the whole interval of its definition for any values θ0 . As an example, we give graphs of functions fθ , f1θ in the interval -80°θ80° . Coefficients a1 , b1 are calculated at a value θ0=60° .

The solid curve in Figure 2 is a graph of the function fθ , the dot curve is a graph of the function f1θ. The graphs practically coincide. Hence, the function f1θ can be used instead of the function fθ in calculating the integrals (3).

Figure 2 Approximation of the function fθ

Now the quadratures (3) are integrated in elementary functions In calculating the integrals we take t0=x0=y0=0 . We integrate the first of the integrals (3). For the coordinate x we obtain:

x=x0-1gθ0θV2dθ==A1lntanθ-b2tan2θ+btanθ+c-A2arctan2tanθ+bΔθ0θ.

Here we introduce the following notation:

a=1kV02cos2θ0+fθ0,d0=-ab1,d1=a1b1,p1=-d02+d024+d13273,p2=-d13p1,b=p1+p2,c=d1+b2,Δ=b2-4c,A1=12gkb1c+b2,A2=6bA1Δ,F1θ=A1lntanθ-b2tan2θ+btanθ+c-A2arctan2tanθ+bΔ.

Thus, the dependence xθ has the following form:

xθ=F1θ-F1θ0. (5)

We integrate the second of the integrals (3). For the coordinate y we obtain:

y=y0-1gθ0θV2tanθdθ==bF1θ+A3arctan2tanθ+bΔθ0θ.

Here we introduce the following notation:

F2θ=bF1θ+A3arctan2tanθ+bΔ,A3=2gkb1Δ.

Thus, the dependence yθ has the following form:

yθ=F2θ-F2θ0. (6)

Consequently, the basic functional dependencies of the problem xθ,yθ are written in terms of elementary functions. The main characteristics of the projectile's motion are the following (Fig. 1):

H - the maximum height of ascent of the projectile,

T - motion time,

L - flight range,

xa - the abscissa of the trajectory apex,

ta - the time of ascent,

θ1 - impact angle with respect to the horizontal . Using formulas (5) - (6), we find:

xa=x(0),H=y(0). (7)

The third integral (3) is not 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 θ1 is determined from the condition yθ1=0 . Then we have

L=xθ1,T=22Hg,ta=T-kHVa2,
Va=V0cosθ01+kV02cos2θ0fθ0. (8)

We note that formulas (5) - (6) also define the dependence y=yx in a parametric way.

4. The results of the calculations. Field of application of the obtained solutions

Proposed formulas have a wide region of application. 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. As calculations show, trajectory of the projectile y=yx and the main characteristics of the motion L , H , xa have accuracy to within 1% for values of the launch angle and for the parameter p within ranges

0 ° < θ0 <90 ° , 0<p 60.

Figure 3 presents the results of plotting the projectile trajectories with the aid of formulas (5) - (6) over a wide range of the change of the initial angle θ0 with the following values of the parameters

V0=80m/s,k=0.000625s2m2,g=9.81m/s2,p=4.

Figure 3 The graphs of the trajectory y=y(x) at launching angles θ0= 20 ° , 45 ° , 80 °

The used value of the parameter k is the typical value of the baseball drag coefficient.

Analytical solutions are shown in Fig. 3 by dotted lines. The thick solid lines in Fig. 3 are obtained by numerical integration of system (1) with the aid of the 4-th order Runge-Kutta method. As it can be seen from Fig. 3, the analytical solutions (dotted lines) and a numerical solutions are the same.

Figure 4 represents the results of plotting the projectile trajectories with the aid of formulas (5) - (6) over a wide range of the change of the initial velocity V0 . In this case the values of the parameter p vary from 1 to 9.

Figure 4 The graphs of the trajectory y=y(x) under the initial conditions θ0= 40 ° , V0= 40 m/s, 80 m/s, 120 m/s. 

As an example of a specific calculation using formulas (5) - (6), 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=0.22s2m2,V0=50m/s,θ0=40°,p=55

The trajectory of the shuttlecock is shown in Fig. 5. The second column of Table 1 contains range values calculated analytically with formulae (7) - (8). 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 in calculating the basic motion parameters L , H , xa does not exceed 1 %. The parameters T , ta are determined in this example with low accuracy due to the large value of the parameter p . With smaller values of the parameter p , the values of T , ta are calculated rather accurately. For example, for p= 4, the errors in calculating these parameters do not exceed 1.5 %.

Figure 5 The trajectory of the shuttlecock. 

Table 1 Basic parameters of the shuttlecock movement. 

Parameter Analytical Numerical Error
value value (%)
L , (m) 11.27 11.34 -0.6
H , (m) 5.10 5.06 0.8
T , (s) 2.04 1.93 5.7
xa , (m) 7.91 7.84 0.9
ta , (s) 0.66 0.71 -7.0
θ1 -82.2 ° -79.4 ° 3.5

Thus, a successful approximation of the function fθ 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.

5. Conclusions

The proposed approach based on the use of analytic formulas 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 formulas containing elementary functions Moreover, numerical values of the sought variables are determined with high accuracy in a wide range of physical parameters. It can be implemented even on a standard calculator.

Of course, the proposed approach does not replace the direct numerical integration of the equations of the projectile motion, but only supplements it. The value and the advantage of the proposed formulas are that they replace a large number of approximate analytical solutions obtained previously by other authors. Thus, proposed formulas make it possible to study projectile motion with quadratic drag force even for first-year undergraduates.

References

[1] M. Turkyilmazoglu, European Journal of Physics 37, 035001 (2016). [ Links ]

[2] C. Belgacem, European Journal of Physics 35, 055025 (2014). [ Links ]

[3] A. Vial, European Journal of Physics 28, 657 (2007). [ Links ]

[4] G.W. Parker, American Journal of Physics 45, 606 (1977). [ Links ]

[5] K. Yabushita, M. Yamashita and K. Tsuboi, Journal of Physics A: Mathematical and Theoretical 40, 8403 (2007). [ Links ]

[6] P. Chudinov International Journal of Nonlinear Sciences and Numerical Simulation 3 121 (2002) [ Links ]

[7] P. Chudinov Revista Brasileira de Ensino de Física 35, 1310 (2013). [ Links ]

[8] P. Chudinov, European Journal of Physics 25, 73 (2004). [ Links ]

[9] B.N. Okunev, Ballistics (Voyenizdat, Ìoscow, 1943). [ Links ]

[10] S. Timoshenko and D.H. Young, Advanced Dynamics (McGrow-Hill Book Company, New York, 1948). [ Links ]

Received: May 01, 2017; Revised: July 03, 2017; Accepted: July 12, 2017

*Correspondence email address: chupet@mail.ru.

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