Massive stars ending their lives in supernova explosions...

Massive stars ending their lives in supernova explosions produce the nuclei of all the atoms in the bottom half of the periodic table by fusion of smaller nuclei. This problem roughly models that process. A particle of mass $m$ moving along the $x$ axis with a velocity component $+u$ collides headon and sticks to a particle of mass $m / 3$ moving along the $x$ axis with the velocity component $-u .$ (a) What is the mass $M$ of the resulting particle? (b) Evaluate the expression from part (a) in the limit $u \rightarrow 0 .$ (c) Explain whether the result agrees with what you should expect from nonrelativistic physics.

Key Concepts

Conservation of Energy and Momentum in Relativity

In relativistic collisions, both energy and momentum must be conserved, but they are linked through the relativistic expressions, which account for the contributions of both rest mass and kinetic energy. This conservation governs the outcome of collision problems by ensuring that the derived expressions for final mass and velocity satisfy the relativistic formulations rather than their classical counterparts.

Nonrelativistic Limit

The nonrelativistic limit of a relativistic expression is obtained when the velocities involved are much less than the speed of light, reducing the relativistic formulas to their classical forms. This principle ensures that the relativistic approach aligns with the intuitive, classical results when the kinetic energies are low enough that the contributions from mass–energy equivalence become negligible.

Relativistic Inelastic Collisions

In relativistic inelastic collisions, colliding objects stick together and merge, so while energy and momentum are conserved, the relationship between mass and energy (via E=mc²) makes the final mass different from the simple arithmetic sum of the initial masses. This is because kinetic energy contributes to the total energy content of the system and thus to its mass in a relativistic context.

Energy–Mass Equivalence

Energy–mass equivalence, encapsulated in the formula E=mc², is a core principle of relativity stating that energy has mass and vice versa. In collisions, not only does rest mass contribute to the total energy, but kinetic energy does as well, so the fusion process increases the effective mass of the final product beyond the classical sum of the colliding particles' rest masses.

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