In a parallel universe, a spaceship is traveling from Earth to a star that is 1

Question

In a parallel universe, a spaceship is traveling from Earth to a star that is 1 kiloparsec (kpc) away (1kpc u 3.26 · 103 light years). Each millennium the spaceship covers a distance of .01kpc. However, in each millennium, the universe expands uniformly and the space between the star and the Earth increases by 1kpc. For the purpose of this exercise, you may assume that the expansions happens instantenously at the turn of the millennium1 . For example, in the first millennium, the spaceship travels .01 kpc, so it is only .99 kpc away. Next the universe expands so that the distance between the star and Earth is now 2 kpc, i.e. doubled the previous distance. Hence, the spaceship distance from the 1This is in no way related to how expansion actually works in our universe. HW4-1 star also gets doubled to 1.98 kpc . In the next millennium, the spaceship will travel another .01 kpc to a distance of 1.97 kpc to the star. Then, expansion will cause the star-Earth distance to become 3 kpc, an increase by a factor of 3/2. Hence, the spaceship will be at a distance of 3/2 · 1.97 kpc = 2.955 kpc to the star. You must determine whether the spaceship will ever reach the star. If it will, give an approximation of the number of millennia it will take. [Hint: During millennium i, what fraction of the star-Earth distance does the spaceship travel?]

Explanation / Answer

the space never reach the star because the distan will increases 1kpc per millennium and spaceship travel only 0.01 kpc per millennium/

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