How Far is the Spaceship from the Planet at the Time of Explosion in its Frame

[greendog77]

Homework Statement

A spaceship is approaching a planet at a speed v. Suddenly, the spaceship explodes and releases a sphere of photons traveling outward as seen in the spaceship frame. The explosion occurs in the planet frame when the spaceship is a distance L away from the planet. In the ship's frame, how far is it from the planet at the time of the explosion?

Homework Equations

Lorentz transformations:

x = \gamma(x' - vt)
t = \gamma(t - vx/c^2)

The Attempt at a Solution

I began by directly applying a Lorentz transformation:
x = \gamma(-L - vt) = -\gammaL

This would mean that the planet is a distance \gammaL from the planet in its own frame when it explodes. However, I recalled that to apply a Lorentz transformation, the origins of S' and S must coincide at time t = 0. I'm not quite sure how to apply this requirement to the problem, as the spaceship never reaches the planet. I imagined attaching a long pole to the spaceship such that at the time of explosion in the ship's frame, the end of the pole just reaches the planet (so that something coincides with the planet at time t = 0). Then I got even more confused because the pole reaching the planet and the rocket exploding occurs at different times in the planet frame. Could you guys help me with this problem? Also, is there a way that the Lorentz transformation can be generalized so that it is not necessary for the two coordinate systems to coincide at t = 0? Thanks!

 

[Chestermiller]

Let's assume that the origins of S and S' would have coincided at t=t'= 0 if the spaceship at x' =0 had not exploded when the ship was a distance L from the planet x = 0. So the x location at which the spaceship exploded was x = -L. Since the spaceship was traveling at velocity v, in terms of L and v, at what time t on the clocks in S did the spaceship explode? Using the Lorentz Transformation, at what time t' on the spaceship clock did the explosion take place. As reckoned from the spaceship frame of reference, how far would the planet have moved toward the spaceship during the time interval between t' (at which it exploded) and t' = 0? This is the distance reckoned from the spaceship frame of reference.

Chet

 

[greendog77]

The time t on the clocks in S would have read simply -L/v. The spacetime coordinate of the explosion in the S frame would thus be (-L/v, -L). This means that in S', the coordinate of the explosion would be
(\gamma(-L/v+vL/c^2),\gamma(-L+L)) = (\gamma(-L/v+vL/c^2),0).
This means that the time t' on the spaceship clock is \gamma(-L/v+vL/c^2) at the time of explosion, and the distance that it is from the planet is \gamma(-L+v^2L/c^2). This seems to give me the right answer, thanks so much! Is there a way that the Lorentz transformations can be generalized to fit such a problem?

 

[Chestermiller]

The time t on the clocks in S would have read simply -L/v. The spacetime coordinate of the explosion in the S frame would thus be (-L/v, -L). This means that in S', the coordinate of the explosion would be
(\gamma(-L/v+vL/c^2),\gamma(-L+L)) = (\gamma(-L/v+vL/c^2),0).
This means that the time t' on the spaceship clock is \gamma(-L/v+vL/c^2) at the time of explosion, and the distance that it is from the planet is \gamma(-L+v^2L/c^2). This seems to give me the right answer, thanks so much! Is there a way that the Lorentz transformations can be generalized to fit such a problem?

Well done. I hope you also realize that:

-\gamma(-L+v^2L/c^2)=-L/γ

So the distance, as reckoned from the S' frame of reference, is contracted. Thus, this could have been analyzed as a straight length contraction problem.

Chet