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In this article, I investigated what the maximum height someone can still see a certain space rocket after being launched, from a safe distance, having a criterion the human visual acuity of one minute of arc. After the analitical solution, I apply

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What is the maximum height we can see a certain rocket as an extensive bodywatching a space launch?An intersting interdisciplinary problem.
Neri Luiz von HollebenNovember 1st, 2015
Abstract
In this article, I investigated what the maximum height someone can still see a certain space rocket afterbeing launched, from a safe distance, having a criterion the human visual acuity of one minute of arc. After
the analitical solution, I apply the result to the particular case of the launch of the space rocket Atlas V 401 by
an observer in the safe distance of 3700 meters. This problem has an innovative pedagogical character because
contain in its essence, its main condiction, a restriction of physiological nature, something that the student do
not usually see in the common textbooks of mathmatics. As an example, this problem could be worked physics
and biology class where the methodology of the PBL (problem-based learning) is applied. Many subjects canunfold from this problem as the involvement of students increase, providing, in this way, a valuable theme for
teacher work.
Key words
: visual acuity, visual range, trigonometry, interdiciplinarity, PBL.
Introdution
A precise calculation to determine what the height is
possible someone to see a rocket in a real situation canbe highly complex, becouse we must to know the localweather conditions like a, luminosity, brightness of jet,and so on. The proposal in this article is to know the
range of human vision considering only a normal humam
vision acuity of one minute of arc. This mean that twopoints can not be distinguished when a angle betweenthe points with the vertex in the observer eye can not
be less of one minute of arc according to Fig. (1). This
limit is due the existing physical distance between thephotoreceptor cells of our retina. I propose, therefore,to assign such distinguished points as one belonging torocket tail and other belonging to rocket tip in inicialsituation and, after a certain height from the ground,reach the indistiguishable situation that two points be-
came one point. In other words, when a rocket become apoint, by deﬁnition, we can say that the maximum limit
of human vision resolution is reached.
Figure 1:
Vision acuity.Fonte:
http://canonicalmomentum.tumblr.com/post/ 86342049687/how-to-write-your-name-on-the-moon
1
Results and discussion
The problem consists in to determinate what the ma-ximum height
h
is possible to someone still see a certainspace rocket of high
a
, from a safe distance
D
. I started
by setting the inicial situation as shown in Fig(2a) andthen setting the ﬁnal state with the object in the limit
of a vision as shown in the Fig.(2b).
αDa
(a)
Inicial set.
θ
1
Daθ
2
hα
= 1
′
(b)
Final set in visual acuity limit.
Figure 2:
Inicial set and ﬁnal set of problem.Source: Author.
From Fig.(2a) we obtain the relation of eq.(1):
tan
α
=
aD,
(1)and from Fig.(2b):
tan
θ
1
=
hD,
(2)
tan
θ
2
=
h
+
aD .
(3)To solve we use the main restriction of problem:
θ
2
−
θ
1
= 1
.
(4)
In this way, we enploy the trigonometric identity to arcs
diﬀerence.
tan(
θ
2
−
θ
1
) = tan
θ
2
−
tan
θ
1
1 + tan
θ
1
tan
θ
2
.
(5)Replacing (4), (2) e (3) in equation (5), we obtain:
tan1
=
h
+
aD
−
hD
1 +
hDh
+
aD
.
(6)Solving (6) to
h
, gives:
(tan1
)
h
2
+ (tan1
a
)
h
+ (
D
2
tan1
−
aD
) = 0
h
=
a
2
−
4
D
2
−
aD
tan1
−
a
2
,
(7)
where we can despise the negative solution for
h
. Thus,
we have two conditions to the true set of
h
.
a
2
>
4
D
2
−
aD
tan1
a
2
−
4
D
2
−
aD
tan1
> a
However, we can rewrite a solution (7) in function of
initial angle
α
given by equation (1) to give
h
=
1
−
4tan
2
α
+
4tan
α
tan1
−
12
/a .
(8)
In this case, we obtain some conditions for true set
of
h
:
1
>
4tan
2
α
+ 4tan
α
tan1
1
−
4tan
2
α
+ 4tan
α
tan1
>
1
.
(9)
Taking as example the launch of the Atlas V 401
rocket
1
, we obtain the following data:Height
a
= 58
.
3
m (10)Safe distance
D
= 3700
m (11)
Figure 3:
Lauching of the Atlas V 401.Source:
https://en.wikipedia.org/wiki/Atlas_V
Replacing of data (10) and (11) in (7), we obtain:
h
=
58
.
3
2
−
4
3700
2
−
85
.
3 37002
.
9088810
−
4
−
58
.
32
h
= 27
km (12)
1
Local of launch: Kennedy Space Center Launch Complex 39 of NASA, in Merritt Island, Flórida, EUA
2
Conclusion
The result of the eq.(8) is interesting because we could generalize to the case of any object moving away from
us. One can say that it is a description of how the resolution of the human eye varies when objects move away.
From the result obtained in eq.(12), we can conclude that, in principle, we can see clearly the space rocket
Atlas V until it reach 27 km from the ground. This correspond to a second layer of atmospheric
2
, called stratosphere,
that is situated between
10000m
and
50000m
from the ground.
2
Source:
http://www.physicalgeography.net/fundamentals/7b.html
3

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