Articles
Simple and convenient analytical formulas for studying the projectile motion in midair
Peter Chudinov^{1
}^{*
}
Vladimir Eltyshev^{1
}
Yuri Barykin^{1
}
^{1}Engineering 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
mw=mg+R,
where w  acceleration vector of the projectile. Differential equations of the motion, commonly used in ballistics, are as follows [^{9}]
dVdt=gsinθgkV2,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 crosssection 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 wellknown 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+kV02cos2θ0fθ0fθ,fθ=sinθcos2θ+lntanθ2+π4,
(2)
x=x01g∫θ0θV2dθ,y=y01g∫θ0θV2tanθdθ,t=t01g∫θ0θVcosθdθ.
(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 wellknown monograph [^{10}]. The integrals on the righthand 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 secondorder 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 firstorder 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 secondorder 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,f1′θ0=f′θ0.
(4)
From the conditions (4) we find
a1=12cosθ0+3lntan(θ02+π4)2tanθ0,b1=12tan2θ01cosθ0lntan(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°≤θ≤80°
. 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).
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=x01g∫θ0θV2dθ==A1lntanθb2tan2θ+btanθ+cA2arctan2tanθ+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=b24c,A1=12gkb1c+b2,A2=6bA1Δ,F1θ=A1lntanθb2tan2θ+btanθ+cA2arctan2tanθ+bΔ.
Thus, the dependence
xθ
has the following form:
xθ=F1θF1θ0.
(5)
We integrate the second of the integrals (3). For the coordinate
y
we obtain:
y=y01g∫θ0θV2tanθdθ==bF1θ+A3arctan2tanθ+bΔθ0θ.
Here we introduce the following notation:
F2θ=bF1θ+A3arctan2tanθ+bΔ,A3=2gkb1Δ.
Thus, the dependence
yθ
has the following form:
yθ=F2θF2θ0.
(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=TkHVa2,
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=0.000625s2∕m2,g=9.81m/s2,p=4.
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 4th order RungeKutta 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.
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=0.22s2∕m2,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 %.
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 firstyear undergraduates.
References
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