Suppose we are on our way to Alpha Centauri, which is $4\mathrm{.}37$light$-$years away $($in the reference frame of the earth$).$ We are in a space ship which is traveling with a speed of $0\mathrm{.}99712$cc$.$ When we are half$-$way there, we send a signal to both the earth and Alpha Centauri. The signals travel at the speed of light$$cc$.$ In the reference frame of the earth and Alpha Centauri, both signals travel $2\mathrm{.}185$ly and thus arrive at their destinations at the same time.

$($a$)$

In our reference frame, how long does it take for the signal to reach earth? Remember that in our reference frame, the distance between earth and Alpha Centauri is less than $4\mathrm{.}37$ly because of length contraction.

years $(0\mathrm{.}2$ years$)$

$($b$)$

In our reference frame, how long does it take for the signal to reach Alpha Centauri?

years $(0\mathrm{.}002$ years$)$

$($c$)$

These two events $($the signal reaching earth and the signal reaching Alpha Centauri$)$ are simultaneous in the earth's reference frame. How much time elapses between these two events in ourour reference frame?

years $(0\mathrm{.}2$ years$)$