Answer
(2), (1), (3)
Work Step by Step
We were given the rest energy and total energy, respectively ,of three particles, expressed in terms
of a basic amount A as (1) A, 2A; (2) A, 3A; (3) 3A, 4A.
Without calculations, we want to rank them according to their Lorentz; greatest first.
The total energy, E, is the product of the Lorentz factor, $\gamma$, and the rest energy, $mc^2$. We can arrange it to isolate the Lorentz factor.
$$E = \gamma mc^2$$
$$\gamma=\frac{E}{mc^2} $$
We can see that the Lorentz factor is the ratio of total energy to rest energy.
Particle (2) has the largest ratio at 3. Particle (1) has the next largest ratio at 2. Particle (3) has the smallest ratio at 4/3.
