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?
between 300,000 and 400,000 km/s
200,000 km/s
400,000 km/s
between 200,000 and 300,000 km/s
To determine how fast an observer in one of the galaxies would see the other galaxy moving away, we need to consider the concept of the relativistic Doppler effect.
The relativistic Doppler effect describes how the frequency and wavelength of light are affected by the relative motion between the source emitting the light and the observer measuring it. In the case of two galaxies moving away from each other, the light waves from one galaxy will be stretched or redshifted.
The formula for the relativistic Doppler effect is given by:
v' = (v + v₀) / (1 + (v × v₀) / (c²))
Where:
v' is the observed velocity of the second galaxy relative to the observer in the first galaxy,
v is the velocity of the first galaxy relative to the observer on Earth (200,000 km/s),
v₀ is the velocity of the second galaxy relative to the first galaxy (200,000 km/s), and
c is the speed of light in a vacuum (299,792 km/s).
Substituting the given values into the formula:
v' = (200,000 km/s + 200,000 km/s) / (1 + (200,000 km/s × 200,000 km/s) / (299,792 km/s)²)
Evaluating the expression:
v' = (400,000 km/s) / (1 + (40,000,000 km²/s²) / (89,875,517,184 km²/s²))
v' = (400,000 km/s) / (1 + 0.445)
v' = (400,000 km/s) / (1.445)
v' ≈ 276,669 km/s
Therefore, an observer in one of the galaxies would see the other galaxy moving away at a speed of approximately 276,669 km/s, which is between 200,000 and 300,000 km/s.