To calculate the age of the twin brother when the astronaut returns to Earth, we can use the concept of time dilation from special relativity. According to time dilation, time appears to pass more slowly for an object in motion relative to an observer at rest.
The formula for time dilation is:
t' = t / √(1 - v^2/c^2)
Where: t' is the time experienced by the moving astronaut (ship time). t is the time experienced by the observer on Earth (Earth time). v is the velocity of the astronaut relative to Earth (0.75c, or 0.75 times the speed of light). c is the speed of light.
Let's calculate the time experienced by the astronaut during the roundtrip:
t' = 25 years (ship time)
Now we can plug in the values and solve for t:
25 = t / √(1 - (0.75c)^2/c^2) 25 = t / √(1 - 0.5625) 25 = t / √(0.4375) 25 = t / 0.6614
t = 25 * 0.6614 t ≈ 16.535 years (Earth time)
Therefore, the astronaut would experience a time of approximately 16.535 years during the roundtrip. The difference between the twin brother's age when the astronaut left and when he returns is:
Age difference = 25 years (twin brother's age when astronaut left) - 16.535 years (time experienced by the astronaut) Age difference ≈ 8.465 years
So, the twin brother would be approximately 8.465 years older when the astronaut returns to Earth.