Journal of Applied Mathematics and Physics, 2016, *, *-*
http://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
Derivation of Specific Velocity of Body Moving
Under Gravity with Zero Total Energy*
T.V.B.S.Satyanarayana Murthy
(Affiliation): Civil Design Department of Power Projects Division, BGR Energy Systems Limited, Chennai, India
Email: [email protected]
Abstract
How to cite this paper: Author 1, Author 2
and Author 3 (2016) Paper Title. ******, *,
*-*.
http://dx.doi.org/10.4236/***.2016.*****
In this paper, the author shows that when a body moves under the influence of gravity,
due the gravitational effect of a gravitating body, there exists a condition such that the
total energy of the body is zero when the body moves at specific velocity derived in
this paper. The moving body will have positive total energy, once its velocity exceeds a
specific velocity. The author derives the specific velocity from classical gravitation
using concepts of special relativity.
Received: **** **, ***
Accepted: **** **, ***
Published: **** **, ***
Copyright © 2016 by author(s) and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Keywords
Classical Gravitation, Special Relativity, Total Energy, Specific Velocity
1. Introduction (Heading 1)
We know that classical gravity leads to total negative energy of a moving body under the
influence of gravity [1]. In this paper, the author applies concepts of special relativity to
classical gravitation and obtains a mathematical formula, which gives a specific velocity
for a moving body. When the body moves at the specific velocity, its total energy is zero.
When the velocity is more than specific velocity the total energy of the body is positive.
2. Formulas in Classical Gravity and Special Relativity (Heading
2)
Let us consider a body with rest mass
m0
be moving at velocity, v under the influence
of gravity of a massive gravitating body. The velocity of moving body satisfies virial
theorem in classical mechanics. (Hereafter we use word velocity to represent orbital velocity as well as the velocity that satisfies virial theorem.) Let the rest mass of the gravi**
Positive total energy in gravity - a possibility
DOI: 10.4236/***.2016.*****
**** **, 2016
Author, Author
tating body be M 0 . Let the classical distance between the two bodies be R . Classical
gravity [1] gives the velocity as
GM 0
R
v
where
G
(1)
is the gravitational constant.
From Equation (1) , we have
v2 
Gravitational potential energy,
GM 0
R
(2)
U of the moving body is given by
U 
GM 0 m0
R
(3)
To consider concepts of special relativity let us define,

where
c
v
c
(4)
is the speed of light in vacuum.
From special relativity [1], we know

where
1
1  2
(5)
 is the Lorenz factor.
From Equation (5), we get
 2  1
1
2
(6)
2.1. Mathematical Treatment (Sub-Heading 2.1)
By rewriting, Equation (6) we get
2 
  1  1
2
(7)
From rest mass of moving body, we have the relativistic mass as
m   m0
Rest mass energy
E0
of the orbiting body given by
E0  m0 c 2
2
(8)
(9)
Author, Author
From Equation (4) and Equation (2) we have
2 
Substituting for
2
GM 0
Rc 2
(10)
from Equation (7) in Equation (10) and rewriting, we get
  2   GM 0 

2 
   1   Rc 
  1  
Multiplying on both sides of Equation (11) with
using Equation
E0
(11)
from Equation (9) and simplifying
we get,
 2 
U
  1 
  1 E0   
(12)
We know that LHS of Equation (12) is the kinetic energy of the orbiting body K given
by
K    1 E0
(13)
Substituting from Equation (13) in, Equation (12) we get
 2 
K  
U
  1 
In low velocity range,
(14)
  1.0 and thus from Equation (14) we get the known classical
result without special relativity.
1
K  U
2
(15)
T U  K
(16)
Total energy is given by
Substituting for K from Equation (14), we get expression for total energy in terms of
U , as
 2 
T U 
U
  1 
(17)
We seek the condition such that the total energy is zero. From Equation (17) we get the
equation in
 .
3
Author, Author
 2   1  0
(18)
By solving Equation (18), we get valid solution for  giving the specific Lorenz factor
 sp
 sp  1.61803398874989
From Equation (19) and Equation (6) we get the specific
 sp  0.786151377757423
(19)
 sp
as
(20)
From Equation (20) and Equation (4) we get the specific velocity, vsp , as
vsp  0.786151377757423c
(21)
We note that the specific velocity is a constant. Up to specific velocity, the total energy of
a moving body is negative, at specific velocity, its total energy is zero and beyond specific
velocity, its total energy is positive.
From the specific velocity and Equation (2), we can get the specific limiting distance,
Rsp , below which the total energy is positive for a given gravitating mass.
GM
Rsp  2 0
vsp
(22)
From the specific velocity and Equation (2), we can get the specific limiting gravitating
mass, M 0 sp , above which the total energy is positive for a given distance.
M 0 sp 
Rvsp2
G
(23)
3. Conclusion
The mathematical equation derived with the help of concepts of special relativity, for
body moving towards its gravitating body indicates the existence of specific velocity beyond which the moving body will have positive total energy. The specific velocity is a
constant.
Acknowledgements
The author gratefully acknowledges the permission granted by the management of M/s
BGR Energy Systems Limited, Chennai, India to publish this work.
References
[1]
4
Walker, J. (2011) Halliday/Resnick Fundamentals of Physics. 8th Edition, John Wiley &
Sons (Asia) Pte. Ltd. Wiley India Pvt. Ltd. , New Delhi.