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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