SHOW WORK PLEASE!!! StarChip, 0,x 0 4 light years Earth, -0,0 Proxima b lan- A S

Question

SHOW WORK PLEASE!!!

StarChip, 0,x 0 4 light years Earth, -0,0 Proxima b lan- A StarChip (interesting reading: http://www.sciencealert.com/this-russian-billionai to-travel-to-proxima-b-our-closest-earth-like-planet) travels at a constant velocity v-0.8c to a planet called Proxima b (earth 2.0") which is 4 light years away from the earth. Both the earth and the planet are assumed to be still, their clocks are synchronized, and their inertial frame is S. The StarChip's inertial frame is S (1) In S frame, how long will the StarChip take to reach Proxima b? (2) In S' frame, how long will the spaceship take to reach Proxima b (using time dilation)? (3) If you (an ant-man) are in the StarChip (S' frame), you see the earth moving away from you and the planet moving towards you. We can treat S' frame at rest and S frame moves at a constant velocity-v. Find the distance L' between the earth and the planet in S' frame (using length contraction). In S' frame, how long the planet will take to 'reach' the StarChip? Using time dilation, in S frame, how long the planet will take to 'reach' the StarChip (is this number the same with that in question (1))?

Explanation / Answer

given, speed of spaceship, v = 0.8 c

distance between the planet and earth, d = 4 light years

a. in S frame (attached to earth) time taken to reach the planet = t

using v = d/t

t = d/v = 4 LY/0.8 c = 5 years

b. in S' frame, time taken for this journey = t'

t' = t*sqroot(1 - (v/c)^2) = 5(1 - (0.8)^2) = 3 years

c. distance between earth and planet from frame reference of the ship, L' = d*sqroot(1 - (v/c)^2) = 4*sqroot(1-0.8^2) = 2.4 ly

D. time difference as measured from earth for round trip = 2*(t - t') = 2*(5 - 3) = 4 years

so after the round trip, the twin brother that stayed on earth will be 4 years older to the brother who went on woth the journey

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