Final answer:
To calculate the rest energy, the Lorentz factor is used to first solve for the rest mass using the given momentum and velocity. Afterward, the rest mass is converted to energy using Einstein's mass-energy equivalence principle, yielding a rest energy of approximately 588.7 KeV.
Step-by-step explanation:
Calculating Rest Energy
The rest energy of a particle can be determined by using the energy-momentum relationship in special relativity given by the equation:
E² = (pc)² + (mc²)²
To find the rest energy, we first need to solve this equation for E, where E is the total energy of the particle, p is the momentum, and m is the rest mass.
Given that the momentum p = 5 × 10µ eV/c and the speed v = 0.65c, we can use the relativistic relation between momentum and velocity:
p = γmv
Where γ (gamma) is the Lorentz factor, calculated by:
γ = 1 / √(1 - v²/c²)
Using the given velocity:
γ = 1 / √(1 - (0.65c)²/c²)
≈ 1.32
We can then solve for the rest mass m:
m = p / (γv) = (5 × 10µ eV/c) / (1.32 × 0.65c)
≈ 5.887 × 10µ eV/c²
Now that we have the rest mass, we can find the rest energy by:
mc² = rest energy
Converting mass from eV/c² to kg (1 eV/c² = 1.78 × 10³⁻¹ kg), and then multiplying by c² to get the rest energy:
Rest energy = (5.887 × 10µ eV/c²) × c²
≈ 5.887 × 10µ eV
Finally, since 1 MeV = 10⁶ eV, the rest energy in KeV is:
Rest energy ≈ 588.7 KeV