To find the relative velocity of one galaxy as observed from the other, we should use the relativistic velocity addition formula because their velocities are significant fractions of the speed of light. The formula is given by:
\[ v_{rel} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \]
where \( v_1 \) and \( v_2 \) are the velocities of the two galaxies with respect to a stationary observer (in this case, the Earth), and \( c \) is the speed of light (approximately 300,000 km/s).
Substituting \( v_1 = 200,000 \text{ km/s} \) and \( v_2 = 200,000 \text{ km/s} \):
\[ v_{rel} = \frac{200,000 + 200,000}{1 + \frac{200,000 \times 200,000}{(300,000)^2}} \]
Now, let’s calculate the denominator term:
\[ \frac{200,000 \times 200,000}{(300,000)^2} = \frac{40,000,000,000}{90,000,000,000} = \frac{4}{9} \]
Now, substituting back into the formula:
\[ v_{rel} = \frac{400,000}{1 + \frac{4}{9}} = \frac{400,000}{\frac{13}{9}} = 400,000 \times \frac{9}{13} \approx 276,923 \text{ km/s} \]
So the relative velocity of one galaxy as seen from the other would be approximately 276,923 km/s, which falls under the category of "between 200,000 and 300,000 km/s".
Thus, the correct answer is: between 200,000 and 300,000 km/s.
1 answer