Three enemy spacecraft have been causing trouble in the asteroid belt. They alwa

Question

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.

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 t whereas 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?

Express your answer in meters to three significant figures.

L = 91.5 m

Part B

Now the locals can see that, taking into account relativity, the enemy spacecraft will be in a line that is only 91.5 m long when they're traveling at 90% the speed of light relative to the asteroid. For how long a time period will all three spacecraft be inside of the asteroid?
Express your answer in microseconds to two significant figures.

t = 0.46 s

Part C

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 ttrig will this signal take to get to the rear of the asteroid?
Express your answer in microseconds to two significant figures.
ttrig = 0.72 s

The locals would like the two ends to close simultaneously in their rest frame, so the front trapdoor will be set to spring 0.72s after the first enemy spacecraft passes whereas the rear trapdoor will spring the instant it gets the signal.

Part D

How close drear to the rear of the asteroid will the first enemy spacecraft be when the trapdoors close?
Express your answer in meters to two significant figures.
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Part E

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?
Express your answer in meters to three significant figures.
drear = 21 m

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?

Explanation / Answer

(A) L = 91.5 m

(B) t = 0.46 us

(C) t(trig) = 0.72 us

(D) d(rear) = 21 m

(E) 93.7 m

(F) Diagram 2 was made by the group that thinks that the plan will work; Diagram 1 was made by the other group.

(G) 42 degrees

(H) 42 degrees

(I) The rear door appears to shut first.

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