Answer to Question #290317 in Mechanics | Relativity for Arnica

Question

. If astronauts could travel at v = 0.950c, we on Earth would say it takes (4.20/0.950) = 4.42 years to reach Alpha Centauri, 4.20 light-years away. The astronauts disagree.

    (a) How much time passes on the astronauts’ clocks?

    (b) What is the distance to Alpha Centauri as measured by the astronauts?

Answer 1

(a) Time will be dilated by a factor of γ in astronaut's frame.

Time

$t = t_0 \times γ \\ γ = \sqrt{1 - (\frac{v}{c})^2 } \\ = \sqrt{1 - (\frac{0.950c }{ c})^2 } \\ = \sqrt{0.0975} \\ = 0.312$

time as measured by astronaut $= 4.42 \times 0.312 = 1.379 \; years$

(b) Distance measured $= L_0 \times γ = 4.2 \times 0.312 = 1.31 \; light \;years$