A spaceship is observed from the Earth to be moving toward a star at a speed of 0.80 c. If the distance from the spaceship to the star is 1.6 ly as measured from Earth, how much time passes on the spaceship while traveling to the star?
A) 3.3 y
B) 2.0 y
C) 1.3 y
D) 1.2 y
E) 0.96 y

Question

Quizplus · Accepted Answer

The time that passes on the spaceship can be calculated using the time dilation formula in special relativity, $ t' = \frac{t}{\gamma} $, where $ t $ is the time interval in the Earth frame, $ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $ is the Lorentz factor, and $ v $ is the velocity of the spaceship. First, calculate the time it takes for the spaceship to reach the star from the Earth's frame: $ t = \frac{distance}{speed} = \frac{1.6 \, \text{ly}}{0.80c} = 2 \, \text{years} $. Then, calculate the Lorentz factor: $ \gamma = \frac{1}{\sqrt{1 - (0.80)^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} = \frac{5}{3} $. Finally, calculate the time in the spaceship frame: $ t' = \frac{2 \, \text{years}}{5/3} = 1.2 \, \text{years} $.

A Spaceship Is Observed from the Earth to Be Moving