Final answer:
The answer to whether E ≈ pc when γ = 30.0, as for the astronaut in the twin paradox, is true. The high value of γ significantly reduces the contribution of the rest mass energy, making the approximation E ≈ pc valid.
Step-by-step explanation:
The question involves understanding the relationship between the energy (E), momentum (p), and mass (m) of a relativistic particle such as an astronaut moving at high velocities, as discussed in the twin paradox scenario. It particularly asks whether it is true that E ≈ pc when the Lorentz factor (γ) is equal to 30.0.
The formula (pc)²/(mc²)² = γ² – 1 implies that at large velocities, the momentum term pc is much greater than the mass-energy term mc². Considering the total energy of a relativistic particle is given by E = γmc², and for very high values of γ, such as 30.0, the rest mass energy (mc²) becomes an insignificantly small part of the total energy, making E ≈ pc a valid approximation.
The equation (pc)²/(mc²)² = γ² - 1 represents the relationship between the momentum p and total energy E of an object with mass m, moving at a relativistic speed. When γ = 30.0, we can substitute this value into the equation and solve for E ≈ pc. Plugging in the value of γ, we get (pc)²/(mc²)² = 30.0² - 1 ≈ 900 - 1 = 899. Therefore, the equation E ≈ pc is False when γ = 30.0.
Therefore, the answer is (a) True.