A spaceship makes the long trip from earth to the nearest star system, Alpha Centauri, at a speed of 0.955c. The star is about 4.37 light years from earth, as measured in earth’s frame of reference (1 light year is the distance light travels in a year). (a) How many years does the trip take, according to an observer on earth? (b) How many years does the trip take according to a passenger on the spaceship? (c) How many light years distant is Alpha Centauri from earth, as measured by a passenger on the speeding spacecraft? (Note that, in the ship’s frame of reference, the passengers are at rest, while the space between earth and Alpha Centauri goes rushing past at 0.955c.) (d) Use your answer from part (c) along with the speed of the spacecraft to calculate another answer for part (b). Do your two answers for that part agree? Should they?

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19-12-2022
Mathematics

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A spaceship makes the long trip from earth to the nearest star system, Alpha Centauri, at a speed of 0.955c. The star is about 4.37 light years from earth, as measured in earth’s frame of reference (1 light year is the distance light travels in a year). (a) How many years does the trip take, according to an observer on earth? (b) How many years does the trip take according to a passenger on the spaceship? (c) How many light years distant is Alpha Centauri from earth, as measured by a passenger on the speeding spacecraft? (Note that, in the ship’s frame of reference, the passengers are at rest, while the space between earth and Alpha Centauri goes rushing past at 0.955c.) (d) Use your answer from part (c) along with the speed of the spacecraft to calculate another answer for part (b). Do your two answers for that part agree? Should they?

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debjitbhowal4568 debjitbhowal4568 · Answer 1 · 2022-12-24T12:27:09-05:00

The trip to Alpha Centauri takes 4.37 years according to an observer on Earth, 7.70 years according to a passenger on the spaceship, and 0.53 years based on the distance and speed measured by the passenger. Time dilation affects the perception of time at high speeds.

a) According to an observer on earth, the trip takes 4.37 years.

b) According to a passenger on the spaceship, the trip takes [tex]$\frac{4.37}{\sqrt{1 - (0.955)^2}} = 7.70$[/tex] years.

c) As measured by a passenger on the spaceship, Alpha Centauri is [tex]$4.37 \times \sqrt{1 - (0.955)^2} = 0.51$[/tex] light years distant from earth.

d) Using the speed of the spacecraft and the distance measured by a passenger on the spaceship, the trip takes [tex]$\frac{0.51}{0.955} = 0.53$[/tex] years. These two answers for part (b) do not agree, and this is expected since time dilation affects the perception of time for objects moving at high speeds relative to each other.

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