Newton‘s Second Law of Motion in connection with E=mc² and with the relativistic mass formula provides an alternative derivation of the relativistic energy of the physical body.

Since both the equivalence principle of mass and energy E=mc² and the formula of mass as a function of velocity have been proved without the aid of relativistic axioms, this derivation of relativistic energy represents the third link in the chain of proofs which, starting from classical physics, leads to the Theory of Special Relativity on a simple and intuitive path.

The relativistic energy formula obtained here is used later, together with that of momentum, to demonstrate all the other formulas of Special Relativity, including that of the relativistic addition of velocities.

This last relationship therefore allows the theoretical proof of the constancy of the speed of light.

Description of the proof in reduced form

(For the detailed version, click here).

In the more general case, which also provides for variable masses at high speeds, Newton’s Second Law yields the following differential equation (see here):

\[ dE_k = v^2dm+mvdv \quad\quad (1.5) \]

The relation (1.5) is valid for the infinitesimal increase in the kinetic energy of an unconstrained physical body subject to a constant force in the same direction of motion.

From the relation (1.5) by replacing dm and m with the relations of the mass-energy equivalence principle (6.2) and of relativistic mass (5.4):

\[ dm = \frac{dE_k}{c^2} \quad \quad \quad\quad(6.2)\] \[ m = \frac{m_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad\quad\quad(5.4)\]

We obtain the following differential equation:

\[ dE_k =v^2\frac{dE_k}{c^2}+\frac{m_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}}vdv \quad \]

whose integration provides the expression of relativistic kinetic energy:

\[ E_k = \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}} – m_0c^2\quad\quad (6.4) \]
Alternative derivation of relativistic energy - Kinetic Energy as a Function of Velocity

Since m0c2 corresponds to the energy of the rest mass, from (6.4) it follows that the term:

\[ \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}} = mc^2 \quad \]

is the total energy, equal to the sum of the energy at rest and the kinetic energy of the unconstrained physical body.

This derivation demonstrates a further case of the compatibility of Newton’s law with the Theory of Relativity.

The detailed version of this alternative derivation of relativistic energy is reported in the sixth chapter of the book “Newton and Relativity“.

Continue on the alternative path of relativistic proofs: Alternative proof of the addition of velocities.

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