To determine how fast an observer in one galaxy would see the other galaxy moving away, we need to use the relativistic velocity addition formula, since the speeds involved are a significant fraction of the speed of light (which is approximately 300,000 km/s).
The relativistic formula for adding velocities \( u \) and \( v \) is given by:
\[ w = \frac{u + v}{1 + \frac{uv}{c^2}} \]
Where:
- \( w \) is the relative velocity of one object as seen from the other,
- \( u \) is the velocity of the first object relative to a third reference point (in this case, Earth),
- \( v \) is the velocity of the second object relative to the same reference point (Earth),
- \( c \) is the speed of light.
In this case, both galaxies are moving away from Earth at a velocity of \( 200,000 \) km/s.
Let's use \( u = 200,000 \) km/s and \( v = 200,000 \) km/s.
Now substituting these values in the formula, we first need \( c^2 \):
\[ c^2 = (300,000)^2 \]
Now we can put the values into the relativistic addition formula:
\[ w = \frac{200,000 + 200,000}{1 + \frac{(200,000)(200,000)}{(300,000)^2}} \]
Calculating the denominator:
\[ \frac{(200,000)(200,000)}{(300,000)^2} = \frac{40,000,000,000}{90,000,000,000} = \frac{4}{9} \]
Now substitute this back into the equation for \( w \):
\[ w = \frac{400,000}{1 + \frac{4}{9}} = \frac{400,000}{\frac{13}{9}} = 400,000 \times \frac{9}{13} \approx 276,923 \text{ km/s} \]
Thus, the velocity of one galaxy as seen from the other galaxy is approximately \( 276,923 \) km/s.
So, the answer would be between 200,000 and 300,000 km/s.