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: Identify the velocities

Let the velocity of one galaxy relative to Earth be $v_1 = 200000$ km/s and the velocity of the other galaxy relative to Earth be $v_2=200000$ km/s.

Step2: Use the relativistic velocity - addition formula

In special relativity, for velocities close to the speed of light (the speed of light $c = 299792$ km/s, and these velocities are significant fractions of $c$), the formula for relative velocity $u$ of two objects moving in opposite directions with velocities $v_1$ and $v_2$ relative to a third - frame (Earth in this case) is $u=\frac{v_1 + v_2}{1+\frac{v_1v_2}{c^{2}}}$. Substituting $v_1 = 200000$ km/s and $v_2 = 200000$ km/s and $c = 299792$ km/s into the formula:
\[

$$\begin{align*} u&=\frac{200000 + 200000}{1+\frac{200000\times200000}{(299792)^{2}}}\\ &=\frac{400000}{1+\frac{4\times10^{10}}{8.98755\times10^{10}}}\\ &=\frac{400000}{1 + 0.445}\\ &=\frac{400000}{1.445}\\ &\approx277093 \end{align*}$$

\]

Answer:

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