If astronauts could travel at v=0.89c, we on Earth would say it takes (4.20/0.890)=4.72 years to reach Alpha Centauri, 4.20 light years away. The astronaut would disagree. (a) How much time passes on the astronaut's clocks? (b) What is the distance to Alpha Centauri according to the astronauts?
β =0.89,
t = tₒ/sqrt(1-β²),
tₒ=t •sqrt(1-β²) = 4.4• sqrt(1-0.89²) = 2 years.
d = v•tₒ=0.89•3•10^8•2•365•24•3600 =1.68•10^16 m
To answer these questions, we need to use special relativity and the Lorentz transformation equations.
(a) Let's start by calculating the time dilation factor, γ, which tells us how time is affected by speed. In special relativity, γ is given by the formula:
γ = 1 / sqrt(1 - (v^2 / c^2))
where v is the velocity of the astronaut and c is the speed of light.
In this case, v = 0.89c, so we can substitute that in:
γ = 1 / sqrt(1 - (0.89c)^2 / c^2)
= 1 / sqrt(1 - 0.89^2)
≈ 2.56
This means that time will appear to pass 2.56 times slower for the astronaut compared to an observer on Earth. So, if it takes 4.72 years according to observers on Earth, we can calculate the astronaut's time using the equation:
time_astronaut = time_earth / γ
= 4.72 years / 2.56
≈ 1.84 years
Therefore, according to the astronauts, only 1.84 years would pass on their clocks during the journey to Alpha Centauri.
(b) To calculate the distance to Alpha Centauri as perceived by the astronauts, we need to use the length contraction formula provided by special relativity. The formula is:
length_contracted = length_rest / γ
In this case, the length_rest is the distance to Alpha Centauri, which is 4.20 light years.
length_contracted = 4.20 light years / γ
= 4.20 light years / 2.56
≈ 1.64 light years
So, according to the astronauts, the distance to Alpha Centauri would appear to be contracted to approximately 1.64 light years.