To find the relative velocity of one galaxy as observed from the other galaxy, we can't simply add the velocities together due to the effects of special relativity. Instead, we use the relativistic velocity addition formula, which is expressed as:
\[ v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]
Where:
- \(v_{rel}\) is the relative velocity,
- \(v_1\) is the velocity of the first galaxy relative to Earth (200,000 km/s),
- \(v_2\) is the velocity of the second galaxy relative to Earth (200,000 km/s),
- \(c\) is the speed of light (approximately 300,000 km/s).
Plugging the values into the formula:
\[ v_{rel} = \frac{200,000 + 200,000}{1 + \frac{200,000 \times 200,000}{(300,000)^2}} \]
Calculating the denominator:
\[ \frac{200,000 \times 200,000}{(300,000)^2} = \frac{40,000,000,000}{90,000,000,000} \approx 0.4444 \]
Thus, the denominator becomes:
\[ 1 + 0.4444 = 1.4444 \]
Now calculating the relative velocity:
\[ v_{rel} = \frac{400,000}{1.4444} \approx 276,000 \text{ km/s} \]
So the observer in one galaxy would see the other galaxy moving away at approximately 276,000 km/s. Therefore, the correct response is:
between 200,000 and 300,000 km/s.
1 answer