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.

 

[asteriatic]

Hi there,

First of all, great job on getting the correct answer with the help of your TA! Let's break down the problem and address your confusion about length contraction.

In this scenario, we have two spaceships passing each other with a relative speed of 0.901 c. This means that from the perspective of each pilot, the other spaceship is moving at a speed of 0.901 c. Now, let's focus on one of the pilots and her perspective.

In her frame of reference, she sees the other spaceship passing by her at a high speed. At the same time, she sets off a flare at the back of her own spaceship. Since she is at rest in her own frame, the flare will travel a distance of 100 m before reaching the front of her spaceship. This is the proper length, Lp, of her spaceship.

Now, let's consider what the other pilot sees. From her perspective, the first pilot's spaceship is also moving at a high speed. This means that the distance between the back and front of the spaceship will appear shorter to her due to length contraction. This is where the equation x'=\gamma(x-vt) comes into play.

The x-axis in this equation represents the distance between the back and front of the spaceship (L), while the x'-axis represents the distance between the back of the spaceship and the point where the flare is set off (x'). The equation takes into account the relative speed (v) between the two pilots and the time (t) at which the flare is set off.

Now, when we plug in the values for v and t, we get a value for x' that is greater than 100 m. This is because the length Lp is contracted due to the high relative speed between the two pilots. However, keep in mind that this is the distance measured by the other pilot, who is moving at a high speed. From her perspective, the back of the first pilot's spaceship appears to be moving away from her at a faster rate, hence the longer distance.

To sum it up, the length of an object appears to be shorter when measured by an observer who is moving at a high speed relative to the object. This is known as length contraction. In this scenario, the length of the spaceship appears to be longer to the other pilot because she is moving at a high speed relative to the spaceship.

I hope this explanation helps clear up your confusion.