To determine the speed that an observer in one galaxy would see the other galaxy moving away, we need to use the relativistic velocity addition formula because the speeds involved are significant fractions of the speed of light.
The relativistic addition of velocities formula is:
\[ v' = \frac{u + v}{1 + \frac{uv}{c^2}} \]
where:
- \(v'\) is the relative velocity of one object as seen from the other,
- \(u\) and \(v\) are the velocities of the two objects (in this case, both are \(200,000 , \text{km/s}\)),
- \(c\) is the speed of light (approximately \(300,000 , \text{km/s}\)).
Substituting \(u = 200,000 , \text{km/s}\) and \(v = 200,000 , \text{km/s}\) into the formula:
\[ v' = \frac{200,000 + 200,000}{1 + \frac{200,000 \times 200,000}{(300,000)^2}} \]
Calculating the denominator:
\[ 1 + \frac{200,000 \times 200,000}{(300,000)^2} = 1 + \frac{40,000,000,000}{90,000,000,000} \approx 1 + 0.4444 \approx 1.4444 \]
Now calculate \(v'\):
\[ v' = \frac{400,000}{1.4444} \approx 276,000 , \text{km/s} \]
This means the observer in one galaxy sees the other galaxy moving away at approximately \(276,000 , \text{km/s}\).
Thus, the correct answer is:
between 200,000 and 300,000 km/s.
1 answer