91Ó°ÊÓ

The energy-momentum relation is a key concept in relativistic dynamics. It connects a particle's energy, momentum, and mass through the equation:

• \( E^2 = (pc)^2 + (m_0c^2)^2 \)

Here, \( E \) stands for the total energy of the particle, \( p \) is its momentum, and \( m_0 \) is the rest mass.
Rest mass energy is represented by \( m_0c^2 \), while \( c \) is the speed of light.
This relation tells us that energy depends not just on mass but also on the momentum a particle carries.
When solving for the rest mass, we rearrange the formula:

• Start with \( E^2 - (pc)^2 = (m_0c^2)^2 \)

This helps find the unknown \( m_0c^2 \) by substituting known values of \( E \) and \( p \). Calculating further, we found \( m_0c^2 = \sqrt{89.44} \), which results in approximately \( 9.46 \, \text{GeV} \).

An important concept while dealing with particles moving at significant fractions of the speed of light is the Lorentz factor \( \gamma \). It is a measure of how much quantities like time, length, and mass transform between two reference frames:

• \( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \)

The Lorentz factor quantifies how time dilates and length contracts as a particle approaches light speed.
For our particle, using its energy and rest mass energy, we derived \( \gamma = \frac{11.2}{9.46} \approx 1.184 \).
This significant value of \( \gamma \) shows the relativistic effects are non-negligible. We can then find the speed, \( v \), using \( \gamma \). By rearranging the Lorentz factor formula, we calculated \( v \approx 0.537c \).

Rest mass energy is a concept that arises from the equation \( E_0 = m_0c^2 \), discovered by Einstein.
This energy is inherent to the particle even when it is at rest, meaning there is no loss of energy when a particle is not moving.
It signifies the energy "contained" due to its mass alone. In our problem, the rest mass energy is calculated as \( 9.46 \, \text{GeV} \).

• This shows the potential energy stored within the mass itself, transformable into kinetic or other energy forms when the particle moves.
• For any moving particle, observing its rest mass energy helps understand both intrinsic energy and relativistic transformations in different frames.

This foundational concept is crucial when examining high-speed particles in accelerators and cosmic phenomena.