Three enemy spacecraft have been causing trouble in the asteroid belt. They always travel in a line, evenly spaced apart, attempting to chase down local spacecraft to steal their goods. The local asteroid colonists have decided to set a trap to capture these three spacecraft. They'll get them to chase one of their fastest ships into an asteroid with a large hole in it and, once the three enemy ships are inside, close two giant trapdoors on each side of the asteroid to catch them. These spacecraft all travel close to the speed of light so the locals will have to take relativity into account. Intelligence about the enemy spacecraft reveals that, in their reference frame, they always travel $90$ m behind their teammate, each spacecraft is $10$ m in length, and their maximum velocity is $90\%$ the speed of light $($relative to the asteroids$).$ The asteroid tunnel is only $215$ m in length. In this problem we will analyze whether the locals will be able to capture the enemy spacecraft after taking into account relativity.

Part A

First, we will construct some of the coordinates in both the asteroid frame and the spacecraft frame. The coordinates for the asteroid frame are given by x$,$ y$,$ z$,$ and twhereas those for the spacecraft frame are given by x$,$ y$,$ z$,$ and t$,$ respectively. Set the time that the front of the first spacecraft enters the asteroid to be t$=$t$=0.$ Also, let the asteroid tunnel be oriented in the x direction so that throughout this problem z$=$z$$ and y$=$y$$ and we can ignore these coordinates.

If the spacecraft are traveling at $90\%$ the speed of light, what is the total length L of the three$-$spacecraft team as observed from the asteroid?