# Velocity, mass, momentum and energy of an accelerated object in relativity

Analytical derivation of relativistic velocity, mass, momentum and kinetic energy of an accelerated object. For Special relativity the momentum of an object of rest mass m0 and velocity u is expressed by equation (1) which is infinite when u equals c. Is it physically meaningful that the momentum of an object becomes infinite while its velocity stays finite? On the other hand, the principle of mass–energy equivalence proposed by Albert Einstein in his article “Does the Inertia of an object Depend Upon Its Energy Content?” has not been rigorously demonstrated, hence it is called a principle not a law. In the contrary, in the theory of Time relativity which is been developed here, momentum and kinetic energy are derived by direct integration and stays limited when u=c.…

The expression of velocity (equation (13)) is directly integrated and thus is mathematically exact. In the contrary, the velocity-addition formula in Special relativity cannot be analytically integrated and one had to make an approximation to compute the velocity of an object which is thus not exact (see section 5.3 of « Introduction to Special Relativity » by James H. Smith).

In Special relativity the expression of relativistic mass is derived with the help of a shock between 2 objects (see section 9.4 of « Introduction to Special Relativity » by James H. Smith). For Time relativity relativistic mass is the derivative of momentum with respect to velocity, which is exactly the definition of mass.

In Special relativity the expression of momentum was derived with the help of a shock between 2 objects (see section 9 of « Introduction to Special Relativity » by James H. Smith) and is infinite when the velocity equals c (see equation (1)). For Time relativity momentum is the integral of infinitesimal change of momentum (see equation (25)). When the velocity of the object equals c its momentum equals the constant π/2 m_0 c, which gives a negative answer to the question of the beginning: “Is it physically meaningful that the momentum of an object becomes infinite while its velocity stays finite? ”

For Time relativity the total kinetic energy of an object is the integral of the work done on it and thus, its expression is mathematically exact. Moreover, when the velocity of the object equals c, its expression equals m0c2 (see equation (45)), which is a proof for the the principle of mass–energy equivalence, while in Special relativity mass–energy equivalence does not has mathematical proof.

At the end, we have derived the momentum-kinetic energy relation for Time relativity, which reduces to the expression of kinetic energy for classical mechanics for small velocity, while the momentum- energy relation in Special relativity does not. In the contrary, this relation is infinite when the velocity equals c.