Astronaut Rachel leaves Earth in a spaceship at a speed of $0.960 c$ relative to an observer on Earth. Rachel's destination is a star system $14.4$ light-years away (one light-year is the distance light travels in one year). According to Rachel, how long does the trip take?
A) 11 years
B) 23 years
C) $4.2$ years
D) 15 years
E) 14 years

Question

Quizplus · Accepted Answer

To find the time according to Rachel, we use the time dilation formula in special relativity, $\Delta t' = \Delta t / \gamma$, where $\Delta t'$ is the time experienced by Rachel, $\Delta t$ is the time in the Earth's frame, and $\gamma$ is the Lorentz factor, given by $1/\sqrt{1 - v^2/c^2}$. First, calculate the time in the Earth frame: $\Delta t = 14.4 \, \text{years}$. Then, calculate $\gamma$ for $v = 0.960c$: $\gamma = 1/\sqrt{1 - (0.960)^2} \approx 3.43$. Thus, $\Delta t' = 14.4 \, \text{years} / 3.43 \approx 4.2 \, \text{years}$.

Astronaut Rachel Leaves Earth in a Spaceship at a Speed 0.960c0.960 c0.960c

Astronaut Rachel Leaves Earth in a Spaceship at a Speed $0.960 c$