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: 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?
Part B: Suppose the trigger is set so that the moment the first enemy spacecraft enters the asteroid a signal is sent at the speed of light to the other end of the asteroid to trigger the rear trapdoor. How long a time will this signal take to get to the rear of the asteroid?
Part C:How close d_rear to the rear of the asteroid will the first enemy spacecraft be when the trapdoors close?
Part D:One of the local students had been studying relativity and began to question the plan. How would this all appear to the enemy spacecraft as they flew through the asteroid? Specifically, if the spacecraft were in an inertial frame (i.e., one with no acceleration) how long would the asteroid tunnel appear to be in their rest frame?