Newtonian Mechanics Solution to E = mc2
Copyright © 2015
Joseph A. Rybczyk
Abstract
Extensive research into the principles of Einstein’s E = mc2 equation led the author to the
alternative E = m0c2 + k equation based on Newtonian mechanics laws of physics. The author’s
equation allows for direct comparison with Einstein’s equation as the mass speed v increases
toward c and thereby provides insights into the meanings of experimental results relating to the
energy equation that were not previously available. Such insights can be useful in broadening
our understanding of the laws of physics that govern the universe and in determining the overall
validity of Einstein’s and the author’s energy equations.
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Newtonian Mechanics Solution to E = mc2
Copyright © 2015 Joseph A. Rybczyk
1. Introduction
Unlike Einstein’s energy equation1 in which the derivation process can be very difficult
to present in a clear and easy to understand manner, the author’s energy equation follows directly
from the laws of Newtonian mechanics. Except for the inclusion of the Lorentz factor2, in order
to complete the derivation process and give the final equation its relativistic qualities, the
author’s equation is based purely on straight forward easy to understand Newtonian principles.
This enhances our ability to understand the underlying meaning of the equation and thereby
assists in our understanding of any pending comparative analyses with Einstein’s equation. The
author did his best to present the findings in a fair and unbiased manner.
2. The Formulas used for the Basic Energy Equation
The nonrelativistic version of the energy equation (i.e., minus the Lorentz factor) is
directly derived from the following four formulas of Newtonian mechanics:
1 2 1
2
3 4 With regard to Eq. (1), W is the work performed by force F over distance d. For Eq. (2), F is the
force causing mass m0 to accelerate at rate a. For Eq. (3), d is the distance traveled at
acceleration rate a during time t. And for Eq. (4), a is the rate of acceleration responsible for
speed v being attained during time t. Note: m0 was used in place of m in Eq. (2) to make it
consistent with the symbol used for rest mass in Einstein’s equation as will be covered in
Sections 4 and 5.
3. Deriving the Newtonian Kinetic Energy Equation
Beginning with Eq. (1) we substitute m0a of Eq. (2) in place of F to obtain
5
for our first step of the derivation process. Then we take the right side of Eq. (3) and substitute it
in place of d in Eq. (5) to obtain
1
2
2
6
that simplifies to
1
2
7
when the acceleration rates a are combined. Next we take v/t of Eq. (4) and substitute it in place
of a in Eq. (7) to obtain
1
2
8
that simplifies to
1
2
9
which is then rearranged into the more familiar form
1
2
10
of the Newtonian kinetic energy equation.
4. Relativizing the Kinetic Energy Equation
To relativized the kinetic energy equation just derived, we need to apply the gamma
version of the Lorentz factor given as
1
11 1
where v is the speed of the mass m0 and c is the speed of light in empty space. This gives
1
1
12
1
in which we can directly factor the rest mass m0 by gamma to obtain
13 where m is the increased mass due to an increase in its speed v according to the principles of
special relativity3. This then gives
3
1
1
14
1
leaving only the use of gamma in the fraction to be explained. In that regard, for v = 0 the
fraction used in equation (14) takes on the familiar value of ½ as demonstrated below.
1
1
1
1
1
1
0
1
1
1
1
√1 0
1
15 2
1
1
1
1
0
For v = c the fraction takes on the less familiar but anticipated form of
1
1
1
1
1
1
1
1
1
1
√1 1
1
1
1
1
0
1
1
1
1
∞
1 16 giving a unit value of 1 in accordance with the principles of relativity. That is, as speed c is
approached, the special relativity speed limit is approached and accordingly the motion of the
mass changes from acceleration to uniform motion. In that case the distance traveled changes
from the ½ value applicable to constant acceleration to the full distance traveled during uniform
motion. With that understood, we can now simplify the relativistic kinetic energy formula to its
simplest form.
By taking Eq. (14) and multiplying the top and bottom of the fraction by gamma we
obtain
17
1
which can be rearranged into the form
1
1
18 or
1
19 or even
4
1
20 or
1
1
21 depending on one’s preference. Note: Eq. (21) was first derived in an earlier work4 by this
author. We are now ready to derive the total energy equation.
5. Deriving the Total Relativistic Energy Equation
Deriving the total relativistic energy equation at this point is a rather simple task. We
need only to add the internal rest mass energy to the relativistic kinetic energy equation just
derived. Since, by definition, the rest mass energy is given by
22 where Erm is the internal energy of mass m0, we simply add this energy to Eqs. (18) through (21).
Thus, we obtain
1
1
23 and
1
24 and
1
25 and
1
1
26 as the complete equation for relativistic energy. All of these forms of the total energy equation
can be given in the simplified form
27 or
5
28 in comparison to the even simpler equation
29 credited to Albert Einstein. His kinetic energy equation is then given as
30 where the kinetic energy is defined as the total energy minus the rest energy of mass m0.
6. Comparing the Author’s Equation’s with Einstein’s
The author’s relativistic forms of the total energy and kinetic energy equations were
directly compared in advanced computer based mathematical programs on many different
occasions and found to give exactly the same results as Einstein’s equations. The only exception
to these findings are that early on Einstein’s kinetic energy equation was found to become
slightly erratic at very low Newtonian velocities. Other than that anomalous behavior, his
formulas give identical results to the author’s formulas out to the 17 decimal place limit of the
most recent mathematical program used. If anything, the resulting comparisons give credibility
to Einstein’s energy equation. It shows that there is supporting theoretical evidence involving
the mechanical principles of physics behind the formulation of the energy equation. As for the
Lorentz gamma factor, however, its use in the above equations does not require a belief in the
special relativity principles of time dilation and distance contraction. An alternate explanation
involves the principle that you cannot accelerate a mass to a faster speed than the speed of the
applied force5. I.e., it doesn’t matter how much the force is increased. The force transferred to
the accelerating mass reduces to zero as the force reaches its inherent speed limit. And, given
that c is the fastest known speed available in the inertial frame in which the force is being applied
to the mass, it is not surprising that speed c cannot be exceeded relative to that frame.
7. Conclusion
Early in his research the author came to the realization that there are two considerations
involved in the application of force5. One involving the magnitude of the force and the other
involving an inherent speed limit at which the force can be applied. For example, the more
locomotives used, one behind the other on the same track, the greater the applied force to that
which is being pushed by the first locomotive. But, regardless of the number of locomotives
employed, the object being pushed cannot be accelerated to a faster speed than the top speed of
the locomotives doing the pushing. This speed limit has an asymptotical nature that according to
experimental evidence6 appears to be represented quite well by the Lorentz gamma factor as used
in this paper. Thus, such use of the gamma factor is not, in itself, an acknowledgment of belief
in the validity of the special relativity principles of time dilation and distance contraction. And
therefore Einstein’s energy equation, E = mc2, as is also the case of the author’s equation, E =
m0c2 + k, does not, by way of proven fact, depend on such principles. And therefore we can
validly claim that the author’s total energy equation along with his kinetic energy equation as
given in this paper can be valid even if the special theory of relativity is not. And, also, as stated
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by the author many times in the past, this does not mean that the speed of light cannot be
exceeded7; only that it cannot be exceeded relative to the inertial frame in which the force is
being applied.
Appendix – Comparison of Kinetic and Total Energy Equations
REFERENCES
1
Albert Einstein, The energy equation is first mentioned in the paper, Does the inertia of a body depend upon its
energy-content? (1905), It is unclear when the equation was first given in the form E = mc2
2
Hendrik Lorentz, First introduced the Lorentz transformation factor in 1899 and 1904. It was later included in
Einstein’s special theory of relativity in 1905.
3
Albert Einstein, the Special Theory of Relativity, originally published under the title, On the Electrodynamics of
Moving Bodies, Annalen der Physik, 17, (1905)
4
Joseph A. Rybczyk, Most Direct Derivation of Relativistic Kinetic Energy Formula, (2010), Eq. (3), Available at
www.mrelativity.net
5
Joseph A. Rybczyk, The Four Principal Kinetic States of Material Bodies, (2005), Breaking the Light Speed
Barrier (2007), Both papers available at www.mrelativity.net
6
Primarily the evidence associated with the acceleration of particles in particle accelerators.
7
Joseph A. Rybczyk, Constant Light Speed – The Greatest Misconception of Modern Science, (2015), Available at
www.mrelativity.net
Newtonian Mechanics Solution to E = mc2
Copyright © 2015
Joseph A. Rybczyk
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in whole or in part in any form
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Appendix – Comparison of Kinetic and Total Energy Equations
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