A particle of mass m whose total energy is twice its rest...

A particle of mass $m$ whose total energy is twice its rest energy collides with an identical particle at rest. If they stick together, what is the mass of the resulting composite particle? What is its velocity?

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Key Concepts

Conservation Laws in Relativistic Collisions

In special relativity, both energy and momentum are conserved in collisions. However, since energy includes both kinetic and rest mass energy (mc²), inelastic collisions where particles stick together can result in a composite object whose rest mass is different from the simple sum of the individual rest masses.

Energy-Momentum Relation

The fundamental relation E² = (pc)² + (mc²)² connects a particle's energy, momentum, and rest mass. This relation is crucial when analyzing relativistic collisions because it allows computation of a particle's momentum from its energy and vice versa.

Invariant Mass

The invariant mass (or rest mass) of a composite system is derived from its total energy and momentum using the formula M = ?(E_total² - (p_total·c)²)/c². This concept is key in relativity because it shows that the rest mass of a bound system is affected by the internal kinetic energy and momentum of its components.

Relativistic Velocity Calculation

In relativistic dynamics, the velocity of a composite particle is determined by the ratio of the total momentum to the total energy (multiplied by the appropriate factors of c). This ensures that the calculated speed complies with the relativistic effects and remains less than the speed of light.

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