A gold nucleus of rest mass 183.473 GeV / c^2 is accelerated from some initial speed to a final

step 1 To calculate the initial speed of the gold nucleus, I will use the formula which is given by V -

A gold nucleus of rest mass 183.473 GeV / c^2 is accelerated...

A gold nucleus of rest mass $183.473 \mathrm{GeV} / \mathrm{c}^{2}$ is accelerated from some initial speed to a final speed of $0.8475 c .$ In this process, $137.782 \mathrm{GeV}$ of work is done on the gold nucleus. What was the initial speed of the gold nucleus as a fraction of $c ?$

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Key Concepts

Relativistic Kinetic Energy

In special relativity, the kinetic energy of an object is given by the expression K = (? - 1)mc², where m is the rest mass and ? (the Lorentz factor) accounts for the effects of traveling at speeds comparable to the speed of light. This formula correctly describes how energy increases nonlinearly with velocity as objects approach relativistic speeds.

Lorentz Factor (?)

The Lorentz factor, defined as ? = 1/?(1 - v²/c²), quantifies the effects of time dilation, length contraction, and the increase in relativistic mass as an object's velocity approaches the speed of light. It is crucial for converting between different energy and momentum expressions in relativistic mechanics.

Work-Energy Principle in Special Relativity

The work-energy theorem in relativity states that the net work done on an object is equal to the change in its relativistic kinetic energy. This principle allows one to relate the energy supplied to an object (through work) to the difference in its kinetic energy, enabling the calculation of variables like initial or final speeds when other quantities are known.

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