momento ANALYTICAL CONSTRUCTION OF THE PROJECTILE MOTION TRAJECTORY IN MIDAIR

Abstract

A classic problem of the motion of a projectile thrown at an angle to the horizon is studied. Air resistance force is taken into account with the use of the quadratic resistance law. An analytic approach is mainly applied for the investigation. Equations of the projectile motion are solved analytically for an arbitrarily large period of time. The constructed analytical solutions are universal, that is, they can be used for any initial conditions of throwing. As a limit case of motion, the vertical asymptote formula is obtained. The value of the vertical asymptote is calculated directly from the initial conditions of motion. 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, a tennis ball, and a shuttlecock of badminton are presented as examples.

Full text

momento Revista de F´ısica, No. 62, Ene - Jun 2021 79
ANALYTICAL CONSTRUCTION OF THE PROJECTILE
MOTION TRAJECTORY IN MIDAIR
CONSTRUCCI ´
ON ANAL´
ITICA DE LA TRAYECTORIA DEL
MOVIMIENTO DE UN PROYECTIL EN EL AIRE
Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Department of Engineering, Perm State Agro-Technological University, 614990, Perm,
Russia
(Recibido: 09/2020. Aceptado: 11/2020)
Abstract
A classic problem of the motion of a projectile thrown at an
angle to the horizon is studied. Air resistance force is taken
into account with the use of the quadratic resistance law. An
analytic approach is mainly applied for the investigation.
Equations of the projectile motion are solved analytically
for an arbitrarily large period of time. The constructed
analytical solutions are universal, that is, they can be used
for any initial conditions of throwing. As a limit case of
motion, the vertical asymptote formula is obtained. The
value of the vertical asymptote is calculated directly from
the initial conditions of motion. 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, a tennis ball, and a shuttlecock of badminton
are presented as examples.
Keywords: Projectile motion, construction of the trajectory, vertical
asymptote.
Resumen
Se estudia un problema cl´asico del movimiento de un
proyectil lanzado formando un ´angulo con el horizonte. Se
tiene en cuenta la fuerza de rozamiento del aire con el uso
Peter Chudinov: chupet@mail.ru doi: 10.15446/mo.n62.90752

80 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
de la ley de resistencia cuadr´atica. En esta investigaci´on
se aplica principalmente una aproximaci´on anal´ıtica. Las
ecuaciones del movimiento del proyectil se resolvieron
anal´ıticamente para un largo periodo de tiempo arbitrario.
Las soluciones anal´ıticas construidas son universales, es
decir, pueden ser utilizadas para condiciones iniciales de
lanzamiento. Como un caso l´ımite de movimiento, se obtiene
la f´ormula de as´ıntota vertical, cuyo valor se calcula
directamente de las condiciones iniciales de movimiento.
No hay necesidad de estudiar el problema num´ericamente.
Las soluciones anal´ıticas encontradas son altamente precisas
en un amplio rango de par´ametros. El movimiento de una
pelota de baseball, de tenis y un volante de b´adminton se
presentan como ejemplos.
Palabras clave: Movimiento de proyectiles, contrucci´on anal´ıtica de la
trayectoria, as´ıntota vertical.
Introduction
The problem of the motion of a projectile in midair is of
great interest to investigators for centuries. There are a lot of
publications on this problem. It is almost impossible to make a
detailed review of all published articles on this topic. Together
with numerical methods, attempts are still being made to obtain
analytical solutions. Many such solutions of a particular type have
been 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). Both the traditional approaches
and modern methods are used for the construction of analytical
solutions. But these proposed approximate analytical solutions
are rather complicated and inconvenient for educational purposes.
Some approximate solutions use special functions, for example, the
Lambert W function. So, the description of the projectile motion
by means of simple approximate analytical formulas under the
quadratic air resistance has great methodological and educational
importance.

Analytical construction of the projectile motion trajectory in midair 81
The purpose of the present original work is to give simple formulas
for the construction of the trajectory of the projectile motion with
quadratic air resistance for an arbitrarily large period of time.
Numerical studies of projectile motion with quadratic dependence
on projectile speed have been reported in [1–3]. From an educational
point of view, such studies suggest that students are confident in
numerical methods. Therefore, one of the goals of this work is to
give an analytical solution of the trajectory determination problem
as simple as possible from a technical point of view, in order to be
grasped even by first-year undergraduates. It is well known that
the projectile trajectory has a vertical asymptote in the resistant
medium. Construction of the trajectory of the projectile over an
arbitrarily large period of time allows us to determine analytically
one of the important characteristics of the motion - the value
of the vertical asymptote. The importance of this characteristic
of the movement is due to the following circumstances. First,
the approach of the trajectory of the projectile to the vertical is
asymptotically realized, for example, when a badminton shuttlecock
moves. Secondly, this characteristic is important in the case when
the beginning of the projectile trajectory is above the ground
(x0= 0, y0>0, see figure 1). Other characteristics of the projectile
motion are also determined using the proposed analytical formulas.
The subject of research in the proposed paper has something in
common with the content of [4, 5]. In these papers, analytical
formulas were obtained for the value of the vertical asymptote of
the projectile trajectory. However, a significant difference between
the results of this paper and the above-mentioned papers is that
the vertical asymptote formula obtained in this research has
much higher accuracy. In this article, the construction of the
trajectory is carried out using the approach [6]. This approach
allows us 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. From
the point of view of the applied methods of solution, the present
research is a development of the approaches used in [7, 8].

82 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
The conditions of applicability of the quadratic resistance law
are deemed to be fulfilled, i.e. Reynolds number Re lies within
1×103< Re < 2×105.
Equations of projectile motion
Here we state the formulation of the problem and the equations
of the motion [8]. Suppose that the force of gravity affects the
projectile together with the force of air resistance R(see figure
1). Air resistance force is proportional to the square of the velocity
Vof the projectile and is directed opposite the velocity vector.
For the convenience of further calculations, the drag force will be
written as R=mgkV 2. Here mis the mass of the projectile, gis
the acceleration due to gravity, kis the proportionality factor. In
most papers, the motion of the projectile is studied in projections
on the Cartesian axis. Meanwhile, the equations of motion of the
projectile in projections on the natural axes, often used in ballistics
[9], are very useful. They have the following form
dV
dt =−gsin θ−gkV 2,dθ
dt =−gcosθ
V,dx
dt =V cosθ, dy
dt =Vsin θ(1)
Here Vis the velocity of the projectile, θis the angle between the
tangent to the trajectory of the projectile and the horizontal, x, y
are the Cartesian coordinates of the projectile,
k=ρacdS
2mg =1
V2
term
=const,
ρais the air density, cdis the drag factor for a sphere, Sis 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.

Analytical construction of the projectile motion trajectory in midair 83
Figure 1. Basic motion parameters.
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
V(θ) = V0cos θ0
cos θ¿
q1 + kV 2
0cos2θ0(f(θ0)−f(θ))
, f (θ) = sin θ
cos2θ+ ln tan θ
2+π
4(2)
x=x0−1
gZθ
θ0
V2dθ, y =y0−1
gZθ
θ0
V2tan θdθ, t =t0−1
gZθ
θ0
V
cos θdθ (3)
Here V0and θ0are the initial values of the velocity and of the
slope of the trajectory respectively, t0is the initial value of the
time, x0,y0are the initial values of the coordinates of the projectile
(usually accepted t0=x0=y0= 0). The derivation on 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,xand ywe must either integrate system (1) numerically
or evaluate the definite integrals (3). In [8], the integrals were
calculated in elementary functions over most of the angle θ.
The purpose of this study is to calculate the integrals (3) in
elementary functions with the necessary accuracy over the entire
interval of variation of the variable θ: [−π/2≤θ≤θ0].

84 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Obtaining analytical solutions to the problem
To understand this article, we emphasize the following moments. In
[6], a very fruitful idea was proposed for calculating integrals (3).
Based on this idea, in [7, 8] their high-precision analytical solutions
for the variables x,y, and twere obtained over a sufficiently large
interval of variation of the variable θ. In the present work, these
solutions are continued up to the limit value θ=−π/2.
The idea [6] is as follows. The task analysis shows, that equations
(3) are not exactly integrable owing to the complicated nature of
the function (2)
f(θ) = sin θ
cos2θ+ ln tan θ
2+π
4
The odd function f(θ) is defined in the interval −π/2< θ < π/2 .
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. Ref. [6] presents a simple
approximation of a function f(θ) by a second-order polynomial of
the following form
fa(θ) = a1tan θ+b1tan2θ
The function fa(θ) well approximates the function f(θ) only on
the specified interval [0, θ0], since the function faθcontains an even
term. Under the condition, θ < 0, nother approximation is required
because the function f(θ) is odd.
In [7, 8] two approximations of the function f(θ) on the whole
interval −π/2< θ < π/2 were proposed. The first approximation
uses a second-order polynomial, the second approximation uses a
third-order polynomial. In this paper, the first approximation is
used, as a simpler one. Approximation of the function f(θ) by a
second-order polynomial f2(θ) has the following form
f2(θ) = 


α1tan θ+α2tan2θ, on condition θ≥0,
α1tan θ−α2tan2θ, on condition θ≤0
(4)

Analytical construction of the projectile motion trajectory in midair 85
Approximation (4) needs two-parameter, the coefficients α1,
and α2. This approximation also satisfies the condition
f2(0) = f(0) = 0. The coefficients α1and α2can be chosen in
such a way as to smoothly connect the functions f(θ) and f2(θ) to
each other with the help of conditions
f2(θ0) = f(θ0), f0
2(θ0) = f0(θ0) (5)
From conditions (5) we have
α1=2 ln tan(θ0/2 + π/4)
tan θ0
, α2=1
sin θ0−ln tan(θ0/2 + π/4)
tan2θ0
Note that due to the oddness of the functions f(θ) and f2(θ)
equality is held f2(−θ0) = f(−θ0). This function f2(θ) well
approximates the function f(θ) over a sufficiently large interval of
its definition for any value θ0(see figure 3).
Based on approximation (4), in [8] analytical formulas for the
variables x,yandt are obtained by calculating integrals (3). The
results [8] are substantially used in this paper. It is quite necessary
to quote the formulas [8] in this article because without them it
impossible to use the results obtained in this paper. Therefore, we
reproduce the needed formulas from [8]:
x1(θ) = −1
gZθ
θ0
V2dθ =A1arctan 2b2tan θ+ 1
b3
θ
θ0
in case of θ>0,
x2(θ) = −A2arctan 2b2tan θ−1
b4
θ
θ0
in case of θ60,
The following notations are introduced here:
A1=2
gkα1b3
, A2=2
gkα1b4
, b1=1
kV 2
0cos2θ0
+f(θ0)/α1,
b2=α2
α1
, b3=p−1−4b1b2, b4=p−1+4b1b2
Thus, the dependence x(θ) has the following form:
x(θ) =x1(θ)−x1(θ0) in case of θ≥0,
x(θ) =x1(0) −x1(θ0) + x2(θ)−x2(0) in case of θ≤0 (6)

86 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
For the coordinate ywe obtain:
y1(θ) = −1
gZθ
θ0
V2tan θdθ =−B1arctan 1+2b2tan θ
b3+B2ln


−b1+ tan θ+b2tan2θ






θ
θ0
in case of θ≥0,
y2(θ) = −B3arctan −1+2b2tan θ
b4−B2ln



b1−tan θ+b2tan2θ






θ
θ0
in case of θ≤0
The following notations are introduced here:
B1=1
kgα2b3
, B2=1
2kgα2
, B3=1
kgα2b4
Thus, the dependence y(θ) has the following form:
y(θ) = y1(θ)−y1(θ0) in case of θ≥0,
y(θ) = y1(0) −y1(θ0) + y2(θ)−y2(0) in case of θ≤0 (7)
The variable tis defined by the formulas
t1(θ) = 1
g√kα2
arctan "(1 + 2b2tan θ)pb1−tan θ−b2tan2θ
2√b2−b1+ tan θ+b2tan2θ#




θ
θ0
in case of θ≥0
t2(θ) = −1
g√kα2
ln 

−1+2b2tan θ+ 2pb2pb1−tan θ+b2tan2θ




θ
θ0
in case of θ≤0
Thus, the dependence t(θ) has the following form:
t(θ) = t1(θ)−t1(θ0) in case of θ≥0
t(θ) = t1(0) −t1(θ0) + t2(θ)−t2(0) in case of θ≤0 (8)
Consequently, the basic functional dependencies of the problem
x(θ), y(θ), t(θ) are written in terms of elementary functions.
We note that formulas (6) – (8) also define the dependencies
y=y(x), y =y(t), x =x(t) in a parametric way. The obtained
formulas very well describe the trajectory of the projectile in the
range of variation of the angle θ, corresponding to the condition

Analytical construction of the projectile motion trajectory in midair 87
Figure 2. The graphs of the trajectory y= y(x) at launching angles θ0= 20o,
50o, 80o, V0= 80 m/s.
y≥0. An example of the use of formulas (6) – (8) is shown in
figure 2.
Figure 2 presents the results of plotting the projectile trajectories
with the aid of formulas (6) – (8) over a wide range of the change
of the initial angle θ0. The following parameters values are used
k= 0.000625s2/m2, g = 9.81m/s2
The used value of the parameter kis the typical value of the
baseball drag coefficient [4]. The thick solid lines in figure 2 are
obtained by numerical integration of system (1) with the aid of the
4-th order Runge-Kutta method. The red dots lines are obtained
using analytical formulas (6) – (8). As it can be seen from figure 2,
formulas (6) – (8) with high accuracy approximate the trajectory
of the projectile in a fairly wide range of angle θ, satisfying the
condition y≥0.
Analytical solutions for the approximation function f(θ) as
the trajectory approaches the asymptote
Thus, analytical solutions are constructed in [8] for the projectile
coordinates x,yfor any values of the angle of inclination of the
trajectory θ. However, at θ→ −π/2, the numerical and analytical
solutions begin to diverge. This can be seen even in figure 2. In

88 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
order to prevent this, we proceed as follows. We divide the full
interval of the change in the angle θof the slope of the trajectory
(−π/2≤θ≤θ0] into two intervals:
(−π/2≤θ≤θ1]and [θ1≤θ≤θ0]
We choose the value θ1taking the following considerations into
account. Since the odd function f2(θ) approximates well the
trajectory of the projectile in the segment [−θ0≤θ≤θ0] and even
further, we take the middle of the gap (−π/2≤θ≤ −θ0] as the
value θ1. In that way θ1=−θ0/2−π/4. Now we will approximate
the function f(θ) on the interval (−π/2≤θ≤θ1] by the function
f3(θ) = β0+β1tan θ−β2tan2θ
In contrast to the two-parameter approximation (4), this
approximation is three-parameter. The presence of the third
parameter β0allows us to impose an additional condition on the
function f3(θ) in addition to the conditions (5). We impose the
following conditions on the function f3(θ)
f3(θ1) = f(θ1), f 0
3(θ1) = f0(θ1), f3(−89◦) = f(−89◦) = −3287.381 (9)
The first two conditions are similar to conditions (5), the third
condition ensures that the trajectory approaches the asymptote.
From conditions (9) we find the coefficients β0, β1, β2:
β2=(f(θ1) + f(89◦)) cos θ1−2 (tan θ1+ tan 89◦)
(tan θ1+ tan 89◦)2cos θ1
,
β1=2 (1 + β2sin θ1)
cos θ1
, β0=f(θ1)−β1tan θ1+β2tan2θ1
Figure 3 shows the results of approximating the function f(θ) by
the functions f2(θ) and f3(θ). The solid green line is a graph of
the function f(θ). The red dotted line is a graph of the function
f2(θ) in the gap [θ1≤θ≤θ0]. The black dotted line is a graph of
the function f3(θ) in the gap (−π/2≤θ≤θ1]. As it can be seen
from figure 3, the used functions f2(θ) and f3(θ) approximate well
the function f(θ) over the entire interval (−π/2≤θ≤θ0] . When
plotting the functions f(θ), f2(θ), f3(θ) in figure 3, the initial value
is used θ0= 60◦.

Analytical construction of the projectile motion trajectory in midair 89
Figure 3. Approximation of the function f(θ)by functions f2(θ)and f3(θ)
Now we integrate the first of the integrals (3). For the coordinate
xwe obtain:
x3(θ) = x0−A3arctan 2d1tan θ−1
d2
θ
θ1
,
here
d0=1
β11
kV 2
0cos2θ0
+f(θ0)−β0, d1=β2
β1
, d2=p4d0d1−1, A3=2
gkβ1d2
.
Integrating the second integral (3), we obtain the dependence y(θ):
y3(θ)=y0−


1
gkβ2d2
arctan

2d1 tan θ−1
d2

−1
2gkβ2
ln(d0−tan θ+d1tan2θ)








θ
θ1
The third integral (3) gives the dependence t(θ):
t3(θ) = t0−1
g√kβ2
ln 1−2d1tan θ−2√d1pd0−tan θ+d1tan2θ
θ
θ1
The obtained formulas x3(θ), y3(θ) for the x,ycoordinates allow
to construct analytically the projectile trajectory over the entire
interval of variation of the angle of slope θ. An attempt to use a
four-parameter approximation of a function f(θ) of the form
f4(θ) = β0+β1tan θ−β2tan2θ+β3tan3θ
for solving the problem does not lead to success, since the
integrals (3) cannot be calculated in elementary functions.

90 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Construction of the trajectory. The formula for the vertical
asymptote. The results of the calculations
Now we can analytically construct the trajectory of the projectile
for an arbitrarily long period of time. However, instead of an
unlimited interval [0 ≤t < ∞) for time t, it is more convenient to
use a limited interval (−π/2≤θ≤θ0] for the angle θ.
Thus, the projectile trajectory over the entire interval of the
variation of the angle θ(−π/2≤θ≤θ0] is described by the
equations:
on the interval [0 ≤θ≤θ0]
t(θ) = t1(θ)−t1(θ0),
x(θ) = x1(θ)−x1(θ0),
y(θ) = y1(θ)−y1(θ0);
on the interval [θ1≤θ≤0]
t(θ) = t1(0) −t1(θ0) + t2(θ)−t2(0),
x(θ) = x1(0) −x1(θ0) + x2(θ)−x2(0),
y(θ) = y1(0) −y1(θ0) + y2(θ)−y2(0);
on the interval (−π/2≤θ≤θ1]
t(θ) = t1(0) −t1(θ0) + t2(θ1)−t2(0) + t3(θ)−t3(θ1),
x(θ) = x1(0) −x1(θ0) + x2(θ1)−x2(0) + x3(θ)−x3(θ1),
y(θ) = y1(0) −y1(θ0) + y2(θ1)−y2(0) + y3(θ)−y3(θ1) (10)
In contrast to all previously obtained solutions to the problem of
the movement of a projectile in a medium with a quadratic law of
resistance, formulas (10) allow us to construct the trajectory of the
projectile for any initial conditions of throwing V0, θ0and for any
values of the drag coefficient k. Strictly speaking, this solution is
not a real exact solution. However, the method of approximating
the function f(θ) used to construct the analytical solution gives a
high-precision approximate solution to the problem.

Analytical construction of the projectile motion trajectory in midair 91
An example of dependencies x(θ), y(x) constructed using equations
(10) is shown in figure 4. In the calculations, the following
parameter values were used:
V0= 120 m/s , k = 0.000625 s2/m2, g = 9.81 m/s2, θ0= 45◦.
Figure 4. The graphs of the functions x(θ), y(x).
The thick solid black lines in figure 4 are obtained by numerical
integration of system (1) with the aid of the 4-th order Runge-Kutta
method. The red dots lines are obtained using analytical
formulas (10). As can be seen from figure 4, the analytical solutions
(dotted lines) and numerical solutions are the same over the entire
interval of change of the angle of inclination of the trajectory
(−π/2≤θ≤θ0]. Each of the graphs allows us to approximately
determine the value of the asymptote xas of the projectile trajectory.
The exact value of the asymptote is determined by the expression
xas =x1(0) −x1(θ0) + x2(θ1)−x2(0) + x3(−π/2) −x3(θ1).
Using the previously obtained formulas, we write the final analytical
expression for the asymptote of the projectile trajectory
xas =A1arctan b3
2b1cot θ0−1+A2arctan b4
1−2b1cot θ1+
A3arccot 1−2d1tan θ1
d2(11)

92 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Multipliers A1, A2, A3, and coefficients b1, b3, b4, d1, d2were
introduced earlier. Note that the asymptote value is calculated
directly from the initial conditions of motion V0, θ0,without
integrating the equations of motion of the projectile. In figure
4, the value of the asymptote determined by the numerical
integration of the system of equations of motion of the
projectile (1) is xas = 294.7m. The asymptote is shown by a solid
vertical black line. Formula (11) gives a value of xas = 293.9m.
The error in calculating the asymptote is 0.28 %.
Figure 5. The graph of the function xas(θ0).
Figure 5 shows the dependence xas =xas(θ0).The upper curve
in figure 5 is plotted for parameter values k= 0.000625 s2/m2,
V0= 120 m/s (value of kfor a baseball), bottom curve is plotted
at values k= 0.002 s2/m2, V0= 70 m/s (kvalue for tennis
ball). The middle curve is plotted for shuttlecock of badminton
(k= 0.022 s2/m2, V0= 100 m/s). In this case, the values xas are
increased 10 times. The black dots lines in figure 5 are obtained by
numerical integration of system (1) with the aid of the 4-th order
Runge-Kutta method. The red and green dots lines are obtained
using an analytical formula (11). Black and red (green) curves
completely coincide over the entire interval of variation of the initial
projectile throw angle 0◦≤θ0≤90◦. It should be noted that the
used values of the resistance coefficient kin figure 5 vary 35 times.
This indicates the high accuracy of formula (11) in a wide range of
variation of the parameter k.

Analytical construction of the projectile motion trajectory in midair 93
Comparison of results
Note that an approximate formulas for the vertical asymptote of the
projectile were proposed in [4, 5]. The vertical asymptote formula
in [4] has the following form (in the notations of this article)
xas =cos θ0
gk 
pπ/2/e/V0√k+V0√k
1/V0√k+V0√k
·ln 1 + eV0√k(12)
In contrast to the derived formula (11), this formula has a heuristic
character. Here e= 2.71828, k = 1/V 2
term.The vertical asymptote
formula from [5] is written as follows
xas =L+1
2gk ln "e2V1
V2(cos θ1)V1√k(cos θ2)#(13)
The following designations are introduced in formula (13):
L=VaT, Va=V(0), T = 2s2H
g, H =V2
0sin2θ0
g(2 + kV 2
0sin θ0),
xa=pLH cot θ0, θd=−arctan LH
(L−xa)2, V1=V(θd),
θ2= (θd−π/2) /2, V2=V(θ2)
In this case, the entered parameters L, H, T, Xa, θdhave the
following geometric meaning (see figure 1): H- the maximum height
of ascent of the projectile, xa- the abscissa of the trajectory apex,
L- flight range, T- motion time, θd- impact angle concerning the
horizontal.
The results of comparing the accuracy of formulas (11), (12), (13)
are presented in figures 6, 7, 8. The movement of a baseball, tennis
ball, and a badminton shuttlecock is considered. The values of the
parameters V0and kare taken from [4].
Solid black curves (11) were plotted using formula (11), green and
blue curves (12) were plotted using formula (12), brown curves
(13) were calculated using formula (13). The red dotted curves
are obtained by integrating the system of equations (1) using the

94 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Figure 6. The graphs of the function xas (θ0)of a baseball. The graphs are
constructed for the values V0= 55 m/s, k = 0.000625 s2/m2. The value of the
parameter kcorresponds to the value Vterm = 40m/s.
Figure 7. The graphs of the function xas(θ0)for badminton shuttlecock. The
graphs are constructed for the values V0= 117 m/s, k = 0.022 s2/m2. The
value of the parameter kcorresponds to the value Vterm = 6.7m/s
Runge-Kutta method of the 4th order. From figures 6, 7, 8 it
follows that formula (11) provides the best accuracy. Curves (11)
completely coincide with the numerical solution.
Besides, we note that the obtained formulas (6) - (8) allow us to
determine directly from the initial conditions V0, θ0, in addition
to the asymptote, other important characteristics of the projectile
motion. Then
H=y1(0) −y1(θ0), xa=x1(0) −x1(θ0), ta=t1(0) −t1(θ0)

Analytical construction of the projectile motion trajectory in midair 95
Figure 8. The graphs of the function xas(V0)for shuttlecock of badminton and
tennis ball. The graphs are constructed for the values θ0= 45◦, k = 0.022 s2/m2
for the badminton and k= 0.002 s2/m2for the tennis ball. The top two curves
are for the shuttlecock, the two bottom curves are for the tennis ball. The xas
value is reduced by 10 times for a tennis ball.
These values were previously found analytically in [6].
To determine the characteristics of motion Land T, it is
necessary to find impact angle with respect to the horizontal θd
from the condition y(θd) = 0. T hen L =x(θd), T =t(θd).
Thus, the original results in this research are:
1 - Analytical formulas (10) for constructing the trajectory of
the projectile over the entire interval of variation of the angle of
inclination (−π/2≤θ≤θ0]. The range of applicability of these
formulas is not limited by any conditions (within the limits of
applicability of the quadratic law of resistance of the medium).
The constructed analytical solutions are universal.
2 - Analytical formula (11) for the value of the vertical asymptote
of the projectile trajectory. The resulting formula is superior in
accuracy to all previously proposed formulas.

96 Peter Chudinov, Vladimir Eltyshev and Yuri Barykin
Conclusion
Thus, a successful approximation of the function f(θ) 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 over an arbitrarily large period
of time. A relatively simple analytical for the vertical asymptote
of the projectile trajectory is obtained. 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, the numerical values of the sought variables
are determined with high accuracy. It can be implemented even on a
standard calculator. The proposed analytical formulas can be useful
for all researchers of this classical problem. They can even be people
with high school math skills. Once again, we note the universality
of the proposed solutions, which do not have restrictions on the
initial conditions and parameters.
References
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and C. Christophe, Proc. R. Soc. A. 470, 20130497 (2014).
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[6] M. Turkyilmazoglu, Eur. J. of Phys. 37, 035001 (2016).
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