Okay, let's tackle this problem step by step. So, we have two astronauts with masses 2m and 3m moving at relativistic speeds of 0.6c and 0.5c respectively in the positive x-direction. The question asks for their relative speed and the total energy of the system. The options are A to D.

First, I need to remember the relativistic formulas for velocity addition and total energy. Since they're moving at significant fractions of the speed of light, classical mechanics won't apply here. Let's start with the relative velocity.

Relative velocity in special relativity isn't as straightforward as subtracting the two speeds. I recall the formula for the relative velocity of object B as seen from object A. The formula is (v + u') / (1 + (vu')/c²), where u' is the velocity of the second object in the frame of the first. Wait, but which way are they moving? Both are moving in the same direction, so if one is moving at u and the other at v in the same frame, then the relative velocity would be (v - u)/(1 - vu/c²), but I need to make sure about the signs. Let me confirm the formula.

Yes, when two objects are moving in the same direction, the relative velocity of the second with respect to the first is (v - u) / (1 - (v u)/c²). Here, u is the velocity of the first object, and v is the velocity of the second object in the same frame. So in this case, the first astronaut is moving at 0.6c, and the second at 0.5c. But wait, if they're both moving in the positive x-direction, and the question is about their relative speed, then the second astronaut's speed relative to the first would be (0.5c - 0.6c)/(1 - (0.5c)(0.6c)/c²). Let's compute that.

First, numerator: 0.5c - 0.6c = -0.1c. Denominator: 1 - (0.5c * 0.6c)/c² = 1 - (0.3c²)/c² = 1 - 0.3 = 0.7. So the relative speed is (-0.1c)/0.7 = -0.142857c. Since speed is a magnitude, the relative speed is 0.142857c, which is approximately 0.14c. Looking at the options, option D has v_rel=0.14c, and option A also has 0.14c. So the relative speed is about 0.14c.

Now, for the total energy. The total energy of the system should be the sum of the relativistic energies of each astronaut. The relativistic energy is given by E = γmc², where γ is the Lorentz factor, 1/sqrt(1 - v²/c²).

First, calculate the energy of each astronaut.

For the first astronaut with mass 2m and speed 0.6c:

γ₁ = 1 / sqrt(1 - (0.6c)²/c²) = 1 / sqrt(1 - 0.36) = 1 / sqrt(0.64) = 1 / 0.8 = 1.25

So E₁ = γ₁ * 2m * c² = 1.25 * 2m c² = 2.5 m c².

For the second astronaut with mass 3m and speed 0.5c:

γ₂ = 1 / sqrt(1 - (0.5c)²/c²) = 1 / sqrt(1 - 0.25) = 1 / sqrt(0.75) ≈ 1 / 0.8660 ≈ 1.1547

So E₂ = γ₂ * 3m c² ≈ 1.1547 * 3m c² ≈ 3.4641 m c².

Adding them together: E_total ≈ 2.5 + 3.4641 = 5.9641 m c². Looking at the options, option D has 5.96 mc², which matches this. Option C has 4.96, which is lower, so probably not. Option A has 5 mc², which is close but not exact. Wait, let me check the calculations again, maybe I made a mistake.

Wait, 1.1547 times 3 is 3.4641, yes. 2.5 plus 3.4641 is 5.9641. So approximately 5.96 mc². So the total energy is around 5.96 mc². The relative speed is 0.14c, so that's option D. Let me check the other options to make sure.

Option A: v_rel=0.14c, E=5mc². The total energy here is 5mc², but our calculation is 5.96, so that's lower. Option C: v_rel=0.1c, so that's incorrect because we found 0.14c. So the correct answer should be D.
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