To determine how fast an observer in one galaxy sees the other galaxy moving away, we cannot simply add the velocities because we must take into account the effects of special relativity. The formula for the relativistic addition of velocities is given by:
\[ w' = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \( w' \) is the velocity of one object as seen from the other,
- \( u \) and \( v \) are the velocities of the two objects relative to a common observer (in this case, Earth),
- \( c \) is the speed of light (approximately 300,000 km/s).
Given that both galaxies are moving away from Earth at 200,000 km/s (i.e., \( u = 200,000 \) km/s and \( v = 200,000 \) km/s), we can substitute these values into the equation:
\[ w' = \frac{200,000 + 200,000}{1 + \frac{(200,000)(200,000)}{(300,000)^2}} \]
Calculating the denominator:
\[ (200,000)(200,000) = 40,000,000,000 \] \[ (300,000)^2 = 90,000,000,000 \] \[ \frac{(200,000)(200,000)}{(300,000)^2} = \frac{40,000,000,000}{90,000,000,000} \approx 0.4444 \]
Now calculate the denominator:
\[ 1 + 0.4444 \approx 1.4444 \]
Now we can calculate \( w' \):
\[ w' = \frac{400,000}{1.4444} \approx 276,000 \text{ km/s} \]
Therefore, an observer in one of the galaxies would see the other galaxy moving away at approximately between 200,000 and 300,000 km/s.
The correct response is:
between 200,000 and 300,000 km/s.