Question

two galaxies on opposite ends of the universe are moving away from the earth. each has a velocity of 200,000 km/s relative to the earth. how fast would an observer in one of those galaxies see the other galaxy moving away? (1 point)

between 200,000 and 300,000 km/s
400,000 km/s
200,000 km/s
between 300,000 and 400,000 km/s

Explanation:

Step1: Apply velocity - addition formula in special relativity

The velocity - addition formula for two velocities $u$ and $v$ in the same direction is $u'=\frac{u + v}{1+\frac{uv}{c^{2}}}$, where $c = 300000$ km/s is the speed of light, $u = 200000$ km/s and $v = 200000$ km/s.

Step2: Substitute values into the formula

Substitute $u = 200000$ km/s and $v = 200000$ km/s into the formula:
\[u'=\frac{200000 + 200000}{1+\frac{200000\times200000}{300000^{2}}}\]
\[u'=\frac{400000}{1+\frac{4\times10^{10}}{9\times10^{10}}}=\frac{400000}{1 + \frac{4}{9}}=\frac{400000}{\frac{9 + 4}{9}}=\frac{400000\times9}{13}\approx276923\] km/s

Answer:

A. between 200,000 and 300,000 km/s