Question #290317

. 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?


1
Expert's answer
2022-01-25T06:50:19-0500

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

Time

t=t0×γγ=1(vc)2=1(0.950cc)2=0.0975=0.312t = 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 =4.42×0.312=1.379  years= 4.42 \times 0.312 = 1.379 \; years

(b) Distance measured =L0×γ=4.2×0.312=1.31  light  years= L_0 \times γ = 4.2 \times 0.312 = 1.31 \; light \;years


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