How do you explain the fact that Energy is conserved but the mass doesn't do if they exist a equivalent relation between them and the energy is conserved to all observers?

Is the mass a quantity conserved in a collision elastic relativistic ? "collision is elastic if they leave the same particles entering" so,I want to know if i can write this:

Suppose a rest-mass $m_o$ with collides elastically with other rest-mass $m_o$ stationary

  • Total energy is conserved: $E_1 +E_2 = E_1' + E_2' $ $$ \gamma m_o c^2 = \gamma_1 m_oc^2 + \gamma_2 m_o c^2$$

In relativistic collisions rest mass is conserved, energy is conserved and momentum is conserved. However you have to use the relativistic equations for energy and momentum:

$$\begin{align} E^2 &= p^2c^2 + m^2c^4 \\ p &= \gamma m v \end{align}$$

where $m$ is the rest mass. You appear to be writing the kinetic energy as $\tfrac{1}{2}\gamma m v^2$, but this is not correct.

  • $\begingroup$ Good answer. I may add one general point directed to the title of the question: Rest mass is allways conserved as long as the particle stays the same. Rest mass is an intrinsic, scalar particle property: the rest mass of for example an electron is allways 511keV. The reference frame for that does not matter since the rest mass is Lorentz invariant. The concept of relativistic "mass increase" is controverse and the concept of relative momentum $p=\gamma m v$ is better I think. However in accelerator physics this mass increase picture is still a thing. $\endgroup$ – N0va Aug 19 '16 at 10:59
  • $\begingroup$ Thanks, so, the conservation of energy must be expresed : $E_1^2 +E_2^2= E_1^2'+E_2^2' $ $\endgroup$ – PCat27 Aug 19 '16 at 12:54
  • $\begingroup$ or $$ (E_1 +E_2)^2= (E_1'+E_2')^2 $$ $\endgroup$ – PCat27 Aug 19 '16 at 13:13
  • $\begingroup$ @PCat27: $E_1 +E_2= E_1'+E_2'$ $\endgroup$ – John Rennie Aug 19 '16 at 13:47
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    $\begingroup$ Personally, I wouldn't say "rest mass is conserved". I know where you are coming from on that, but I prefer to emphasize that the (rest) mass of a system can (often does) differ from the sum of the (rest) masses of it's parts. Maybe something on the lines of "the rest masses of individual particles involved in an elastic scattering event are constant (or unaffected by the scattering)". $\endgroup$ – dmckee --- ex-moderator kitten Aug 19 '16 at 18:38

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