To determine how fast an observer in one galaxy sees the other moving away, we must use the principles of relativistic velocity addition rather than simple arithmetic addition, because the velocities involved are a significant fraction of the speed of light.
The formula for relativistic addition of velocities \( u \) and \( v \) is given by:
\[ w = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \( w \) is the resultant velocity as observed from one frame,
- \( u \) and \( v \) are the velocities of the two objects,
- \( c \) is the speed of light (approximately 300,000 km/s).
In this scenario:
- Let \( u = 200,000 \) km/s (velocity of the first galaxy relative to Earth),
- Let \( v = 200,000 \) km/s (velocity of the second galaxy relative to Earth).
Now, we can plug these values into the formula:
\[ w = \frac{200,000 + 200,000}{1 + \frac{(200,000)(200,000)}{(300,000)^2}} \]
Calculating the denominator:
\[ \frac{(200,000)(200,000)}{(300,000)^2} = \frac{40,000,000,000}{90,000,000,000} = \frac{4}{9} \approx 0.444 \]
So the denominator becomes:
\[ 1 + 0.444 \approx 1.444 \]
Now, substitute back to find \( w \):
\[ w = \frac{400,000}{1.444} \approx 277,000 \text{ km/s} \]
Thus, an observer in one galaxy would see the other galaxy moving away at a speed of approximately 277,000 km/s.
Therefore, the correct answer falls into the category of between 200,000 and 300,000 km/s.
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