Two spaceships, each $100$ m long when measured at rest, travel toward each other with speeds of $0\mathrm{.}85$ c as measured by an observer on Earth.

$($a$)$ How fast is each ship traveling as measured by an observer on the other ship?

$($b$)$ How long is each ship as measured by the observer on Earth?

$($c$)$ How long is each ship as measured by an observer on the other ship?

$($d$)$ Now consider two events: event $1$ is when the fronts of the two ships pass each other $($neglect their transverse separation$)$; event $2$ is when the backs of the two ships pass each other. What is the time interval between the two events as measured by an observer on Earth?

$($e$)$ What is the time interval between the events as measured by an observer on one of the ships, taking into account the perceived lengths of the ships by that observer?

$($f$)$ What is the proper time between the two events? Use that proper time to check your answer to part $($e$).$