Spaceship question: Problems with Length Contraction

[harts]

Homework Statement

Two spaceships having rest length 100 m pass each other traveling opposite directions with a relative speed of 0.901 c As the front of the spaceships just cross, each pilot sets off a small flare at the back of her own ship, synchronized to the same instant (t=0 in her own frame).

To each pilot, how far in front does the flare of the back of the other ship occur?

Homework Equations

x'=$\gamma$(x-vt)
L=L_p/$\gamma$

The Attempt at a Solution

I got the correct answer (x'=291 meters) with the help of a TA, but I'm a bit confused by why this is correct. I thought lengths contracted - here the length is extending. My book says that the proper length is measured by an observer for whom the endpoints of the length remain fixed in space. Also it says that L=$\gamma$L_p. If that is the case, then why is the length extending? To me, it seems like x' and L are the same thing in this problem because we are finding how far back the flare occurs.

Any help?

 

[Doc Al]

To me, it seems like x' and L are the same thing in this problem because we are finding how far back the flare occurs.

x' is the distance between the back of the ship at the instant that it fired the flare and the point where the fronts pass. It's not the length of any object.

 

[harts]

x' is the distance between the back of the ship at the instant that it fired the flare and the point where the fronts pass. It's not the length of any object.

Thanks for the reply doc.

I guess I was confused (maybe still confused) about why that distance you just described can't be the length of the ship. Its the distance from the back of the ship to the front.. so the length of the ship.

I know I'm wrong.. I guess I just want an explanation for why my reasoning is wrong. Thanks!

 

[Doc Al]

I guess I was confused (maybe still confused) about why that distance you just described can't be the length of the ship. Its the distance from the back of the ship to the front.. so the length of the ship.

What you need to remember is that simultaneity is frame dependent. The passing of the fronts of the rockets and the firing of the flares are simultaneous only in the frame of each rocket.

Observers in the first rocket will say that the second rocket fired its flare long before the ship noses passed. So that distance does not represent the length of the second rocket, at least according to the first rocket.