Some Geometric Objects Related to a Family of the Ballistic Trajectories
in a Viscous Medium
Zarema S. Seidametova
1 a
and Valerii A. Temnenko
2 b
1
Crimean Engineering and Pedagogical University, 8 Uchebnyi Ln., Simferopol, 95015, Ukraine
2
Independent researcher, Simferopol, Ukraine
Keywords:
Ballistic Trajectories, Linear Resistance, Envelope for a Family of Curves.
Abstract:
Computer geometric modeling is important pre-processing steps in the object’s mathematical representation
using curves that may be constructed using analytic functions, a set of points, or other curves and surfaces. The
paper describes some remarkable curves related to a family of the ballistic trajectories in a viscous medium
with a linear resistance. The envelope of the family of trajectories, the trajectory of the farthest flight and the
curve of maximum flight altitudes are presented in parametric form. A geometric interpretation of the entire
set of ballistic trajectories in the form of some surface (the Galileo’s dome) is also presented.
1 INTRODUCTION
Some classical problems of applied mathematics and
mechanics seem inexhaustible. Each appeal to them
reveals some new facets, highlighting the existence of
hidden connections between various areas of math-
ematics. Galileo’s problem about the motion of a
body thrown at some angle to the horizon was the first
solved problem of dynamics. It was solved by Galileo
long before the appearance of the Newtonian mechan-
ics. The solution is given in his last book “Discorsi
e Dimostrazioni Matematiche Intorno a Due Nuove
Scienze”, published in Leiden in 1638. This book
was translated from Italian and Latin into English by
Henry Crew and Alfonso de Salvio in 1914. Now this
translation is available in the Online Library of Lib-
erty (Galilei, 1914).
“Fourth Day: The motion of projectiles” is the
chapter title of (Galilei, 1914) treating the problem in
the delightful and convincing language of geometry.
This language of the era, perhaps, will seem some-
what heavy to the modern reader. But the epoch had
no other language. Neither Newton’s laws of mechan-
ics nor differential equations existed.
This problem is a traditional and simple task, with
which the study of mechanics and physics often be-
gins. The design of the geometric modeling is widely
used in Computational Fluid Dynamics (CFD) simu-
a
https://orcid.org/0000-0001-7643-6386
b
https://orcid.org/0000-0002-9355-9086
lations. Simple and efficient geometric modeling can
improve the efficiency of flow field simulations for
various applications. Some of the applications de-
scribed in (Bertin, 2017; Zhou et al., 2017; Ma et al.,
2019).
We will consider in this paper some new geomet-
ric objects related to this problem.
In the paper (Seidametova and Temnenko, 2020)
we considered the simplest Galilean version of this
ballistic problem, assuming that only gravity acts on
the flying object. In this paper we examined the bal-
listic problem in a viscous environment. We will as-
sume that, in addition to gravity, a viscous resistance
force
~
F
R
acts on the flying object, which is linearly
dependent on the speed of movement ~v:
~
F
R
= −b~v. (1)
The constant b characterizes the resistance of the
medium. For a physical object at low Reynolds num-
bers, the value b is determined by the well-known G.
G. Stokes formula (Landau and Lifshitz, 1987):
b = 6πaρ
m
ν
m
, (2)
where a is a sphere radius, ρ
m
is a density of the
medium, ν
m
is a kinematic viscosity of the medium.
We take the value of the initial speed of the thrown
body v
0
as a velocity unit, the acceleration of gravity
g as an acceleration unit. With this choice, the unit
of time is
v
0
g
, and the unit of length is
v
2
0
g
. Let t be
the time, x the horizontal coordinate, y the vertical
578
Seidametova, Z. and Temnenko, V.
Some Geometric Objects Related to a Family of the Ballistic Trajectories in a Viscous Medium.
DOI: 10.5220/0011009600003364
In Proceedings of the 1st Symposium on Advances in Educational Technology (AET 2020) - Volume 2, pages 578-583
ISBN: 978-989-758-558-6
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

coordinate (we assume that y ≥ 0), α is the angle that
the initial speed vector makes up with the horizontal
line (0 ≤ α ≤ π/2).
2 FORMULATION OF THE
PROBLEM
Newton’s equations of motion are:
˙v
x
= −βv
x
,
˙v
y
= −1 −βv
y
.
(3)
˙x = v
x
,
˙y = v
y
.
(4)
Here the dot above the letter denotes the time
derivative, v
x
, v
y
are the Cartesian components of the
velocity ~v; β is the dimensionless parameter charac-
terizing the resistance of the medium:
β =
bv
0
mg
. (5)
where m is the mass of a flying object.
If we assume that the flying object is a homoge-
neous sphere of radius a and density ρ
b
, then, taking
into account the Stokes formula (2), the coefficients of
viscous resistance β can be given the following form
β =

9
2
ρ
m
ρ
b

·
ν
m
v
0
ga
2
. (6)
In order for the equations of motion (3) to ade-
quately describe the trajectory, two conditions must
be met:
1. The size of the flying body should be much
smaller than the characteristic dimensions of the
flight path:
v
2
0
ag
 1. (7)
2. The Reynold’s number should be small enough
Re =
v
0
·a
ν
m
 1. (8)
Inequalities (7) and (8) limit the initial velocity
from above and below:
√
ag  v
0

ν
m
a
. (9)
For these constraints to be compatible, the object
must be small enough:
a 

ν
2
m
/g

1/3
. (10)
To prevent inequality (10) from being too burden-
some, experiments with a flying object should be car-
ried out in a medium with a high viscosity, for exam-
ple, in glycerin.
The equations of motion (3) and (4) are supple-
mented by the initial conditions at t = 0:
v
x
(t = 0) = cos α,
v
y
(t = 0) = sin α,
(11)
and
x(t = 0) = 0,
y(t = 0) = 0.
(12)
In the equations of motion (11) α is the departure
angle (the angle that makes the body’s velocity vector
with the axis x at the initial moment). The angle α
obeys the condition:
0 < α ≤
π
2
. (13)
The formulated problem contains one physical pa-
rameter β and one geometric parameter α. Changes of
α in region (13) at fixed β generates a family of bal-
listic trajectories. We investigate in this paper how re-
sistance β affects the appearance of a family of trajec-
tories. We considered the trajectories at y ≥ 0, from
the moment of departure of the object to its fall.
Of particular interest are three curves generated
by the family of trajectories: the envelope of the fam-
ily of trajectories, the trajectory of the farthest flight,
and the locus of the points of maximum flight altitude
when the departure angle changes. In (Seidametova
and Temnenko, 2020) a new composite remarkable
curve was constructed from these three curves, which
we called Galileo’s poleaxe. We will look at how the
parameter β affects these wonderful curves.
3 TRAJECTORIES OF
MOVEMENT
The solutions of the differential equations of motion
(3), (4) with the initial conditions (11), (12) have the
following form:
v
x
= cosα·e
−βt
,
v
y
=
1
β

(1 + β sin α)e
−βt
−1

.
(14)
x =
cosα
β

1 −e
−βt

,
y =

1/β
2


(1 + β sin α)

1 −e
−βt

−βt

.
(15)
Some Geometric Objects Related to a Family of the Ballistic Trajectories in a Viscous Medium
579

Eliminating time t from (15), we can obtain an
explicit equation for the family of ballistic trajectories
in a medium with linear viscous resistance:
y =
1
β
2

(1 + β sin α)
βx
cosα
+ ln

1 −
βx
cosα

.
(16)
4 THE LOCUS OF THE
MAXIMUM LIFTING HEIGHTS
OF THE TRAJECTORIES
At the point of maximum rise of the flying body, the
following condition is met:
v
y
= 0. (17)
Substituting into (17) the expression for v
y
from
(14), we find the flight time t
m
to this point:
t
m
=
1
β
ln(1 + βsinα). (18)
Substituting the value t
m
into the equations of mo-
tion (18), we obtain the equations for the geometric
maximum rise of the trajectory:
x =
1
2
·
sin2α
1 + βsin α
,
y =
1
β
2
(βsin α −ln (1 + β sin α)).
(19)
Relations (19) in a parametric form define the
curve of maximum heights. Figure 1 shows curves
(18) at some values β.
For β → 0 equation (19) yields the equations of
the maximum height curve in the absence of medium
resistance:
x =
1
2
sin2α,
y =
1
2
sin
2
α.
These equations were given in the paper (Sei-
dametova and Temnenko, 2020). These equations de-
scribe the semi-ellipse:

x
1/2

2
+

y −1/4
1/4

2
= 1.
(x ≥ 0;y ≥ 0).
5 THE ENVELOPE FOR A
BALLISTIC TRAJECTORY
FAMILY
The envelope of the family of ballistic trajectories
(16) satisfies the equations of motion (15) and the
condition for the vanishing of the Jacobian
D(x,y)
D(t,α)
:
D(x,y)
D(t,α)
=




˙x ˙y
∂x
∂α
∂y
∂α




= 0 (20)
Relation (20) can be given the form:
v
x
∂y
∂α
−v
y
∂x
∂α
= 0. (21)
Calculating the derivatives by (15)
∂x
∂α
and
∂y
∂α
and
substituting this into (21), we obtain a relation con-
necting the departure angle α and the time t at which
the trajectory touches the envelope:
e
−βt
=
sinα
β + sinα
. (22)
Substitute (22) into equation (15) generates the
envelope equation in parametric form:
x =
cosα
β + sinα
,
y =
1
β
2

β(1 + β sin α)
β + sinα
+ ln

sinα
β + sinα

.
(23)
Since we considered only trajectories with y ≥ 0,
equations (23) describe the section of the envelope
with y ≥ 0 for values α of the parameter satisfying
the inequalities:
α
m
≤ α ≤
π
2
. (24)
where α
m
is the departure angle corresponding to the
trajectory of the maximum flight range. Figure 2
shows the envelope of ballistic trajectories at some
values of β.
6 FLIGHT DISTANCE AND THE
FOLIUM OF GALILEO
The flight range l is the value of the horizontal coordi-
nate x when the vertical coordinate y vanishes. Denote
t
f
the flight time of the object before falling. We also
introduce the notation:
τ = βt
f
. (25)
Assuming in (15) y = 0 we establish a relationship
between the departure angle α and the total flight time
t
f
:
sinα =
1
β
τ −(1 −e
−τ
)
1 −e
−τ
. (26)
Assuming in (15) t = t
f
and substituting t
f
into
the expression for the coordinate x, we find the flight
range l:
l =
1
β
s
1 −

1
β
·
τ −(1 −e
−τ
)
1 −e
−τ

2
·

1 −e
−τ

. (27)
AET 2020 - Symposium on Advances in Educational Technology
580

Figure 1: Curve of maximum heights of ballistic trajectories at a given β.
Figure 2: The envelope of the family of ballistic trajectories for some β.
Figure 3: The folium of Galileo at some values of the dimensionless parameter of viscous resistance β.
Equations (27) together with the relation arising
from (26):
α = arcsin

1
β
·
τ −(1 −e
−τ
)
1 −e
−τ

, (28)
define in a parametric form the dependence of the
flight range l on the departure angle α. The param-
eter of this curve is the value τ.
Figure 3 shows the dependence l = l(α) at some β.
As suggested in (Seidametova and Temnenko, 2020),
this dependence is constructed in the form of a polar
diagram, which we called “The folium of Galileo”.
The flight range l is interpreted as a radial coordinate
in polar coordinates, and the angle α is interpreted as
an azimuthal angle in polar coordinates.
When constructing figure 3, it should be noted that
Some Geometric Objects Related to a Family of the Ballistic Trajectories in a Viscous Medium
581

Figure 4: Galileo’s Poleaxe for a ballistic problem with viscous resistance at some values of the resistance parameter β.
the parameter τ is bounded from above:
τ ≤ τ
∗
, (29)
where τ
∗
is the solution to the equation:
F(τ) =
τ
1 −e
−τ
= 1 + β. (30)
determined by (30) for a given β, we build the folium
of Galileo (27), (28) on the interval of change τ:
0 ≤ τ ≤ τ
∗
. (31)
In the figure 3 ˜x and ˜y some conditional cartesian
coordinates
˜x = l(α) ·cos α,
˜y = l(α) ·sin α.
7 GALILEO’S POLEAXE
Knowing the envelope of the family of ballistic curves
(23) and the curve of maximum altitudes (19), as well
as adding to these curves the trajectory of the farthest
flight, we can build a composite curve – Galileo’s
Poleaxe (figure 4).
When constructing the trajectory of the farthest
flight, it is necessary using curve from figure 3, to set
the angle α
max
corresponding to the farthest flight and
substitute this value of the angle α into equation (15).
8 GALILEO’S DOME
If in the equation of a one-parameter family of the bal-
listic trajectories (16) we reinterpret the triple (x,y, α)
as a triplet of cylindrical coordinates (ρ, z,ϕ): x ≡ ρ;
y ≡ z; α ≡ ϕ, then the equation of the family of curves
(16) turns into the equation of one surface given ex-
plicitly in cylindrical coordinates z = z(ρ, ϕ):
z =
1
β
2

(1 + β sin ϕ)
βρ
cosϕ
+ ln

1 −
βρ
cosϕ

.
(32)
Figure 5: The Galileo’s dome for β = 0.
Figure 6: The Galileo’s dome for β = 2.
It is assumed here that the polar coordinates (ρ,ϕ)
are given in some auxiliary plane (
e
x,
e
y):
e
x = ρcos ϕ;
e
y = ρsin ϕ.
Equation (32) describes (for z ≥ 0 and
AET 2020 - Symposium on Advances in Educational Technology
582

0 ≤ ϕ ≤
π
2
) a certain surface (figure 5), which
we call “Galileo’s dome”. Galileo’s dome provides
a visual representation of the entire set of ballis-
tic trajectories as some whole geometric object
(figure 6).
9 CONCLUSIONS
The paper presents a solution to the problem of a fam-
ily of ballistic trajectories in a medium with linear vis-
cous resistance. The equations of the envelope of the
family of trajectories and the equation of the curve of
the highest elevation of the trajectory are presented in
a parametric form. The polar diagram of the flight
range is presented in parametric form. The paper also
presents a geometric interpretation of the entire set of
ballistic trajectories in the form of the some surface –
the Galileo’s dome.
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