Final answer:
An object whose total energy is twice its rest energy is traveling at approximately 86.6% of the speed of light relative to the observer.
Step-by-step explanation:
The student asked how fast an object is moving if its total energy is twice its rest energy? To answer this, we can use the theory of special relativity. In Einstein's theory, the relationship between the total energy (E), rest energy (E0), and kinetic energy (KE) of an object can be given by the formula:
E = √(E0^2 + (pc)^2)
where p is the momentum of the object and c is the speed of light. Since we know the total energy is twice the rest energy, we can use the relation E = 2E0 and solve for the speed (u) of the object using Einstein's equation for momentum p = γmu, where γ is the Lorentz factor and m is the rest mass of the object.
From the equation E = 2E0, and knowing that E0 = mc^2 and E = γmc^2, we can set up the following equation:
γmc^2 = 2mc^2
which simplifies to γ = 2. The Lorentz factor γ is defined as γ = 1/√(1 - (u/c)^2), and solving for u yields:
u = c √(1 - (1/γ^2))
u = c √ (1 - (1/4))
u = c √(3/4)
u ≈ 0.866c
This means that the object is traveling at approximately 86.6% the speed of light relative to the observer.