An astronaut takes a trip to Sirius, which is located a distance of 8 light-year

Question

An astronaut takes a trip to Sirius, which is located a distance of 8 light-years from the Earth. The astronaut measures the time of the one-way journey to be 1.14 years. If the spaceship moves at a constant speed of 0.99c, how can the 8-ly distance be reconciled with the 1.14-year trip time measured by the astronaut?

What if the astronaut wants to get there in 6.4 years (according to her own clock on the spacecraft), what should be the speed of the spacecraft as measured by the Earth/Sirius? Express your answer in unit of ly/yr. (Since c = 1 ly/yr, the answer in this unit can also be interpreted as the ratio v/c instead.)

Please show work. Thank you.

Explanation / Answer

[1-0.99^2] = 0.141
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As per the observer at earth the time taken by astronaut is 1.14/0.141 = 8.085 years.
The distance he traveled is 8 light years
His velocity = 8 light years/ 8.085 years = 0.9895 light = 0.9895 c
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As per the calcualtions of the astrounaut the earth to Sirius distance is 8*0.141 = 1.128 light years and the time he has taken is 1.14 years to cross this distance
he calculates his speed as 1.128 light years / 1.14 years = 0.9895 light = 0.9895c
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