In their common rest frame, two stars are 90.0 light-years (ly) apart. If they are 12.0 ly apart as measured by the navigator in a spaceship traveling between them, how fast is the spaceship moving? Express your answer in terms of c.
A) 0.991c
B) 0.986c
C) 0.980c
D) 0.972c

Question

Quizplus · Accepted Answer

Let's use the Lorentz transformation to solve the problem:
$$\frac{d}{\sqrt{1-v^2/c^2}} = 12.0\ ly$$
$$\frac{90.0}{\sqrt{1-v^2/c^2}} = 12.0 + vt$$
where $d$ is the distance measured by the spaceship, $v$ is its velocity, and $t$ is the time elapsed according to the navigator. We can solve this system of equations for $v$ and get:
$$v = \frac{d}{t} = c \left(1 - \frac{1}{\sqrt{1+(d/t)^2}}\right) = 0.991c$$
Therefore, the answer is A.

In Their Common Rest Frame, Two Stars Are 90