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) = -$\gamma$L

This would mean that the planet is a distance $\gamma$L 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

 

[Nickie]

I would approach this problem by breaking it down into smaller, more manageable parts. First, I would start by considering the explosion itself and its effects on the spaceship and the photons it releases. The Lorentz transformations can be used to calculate the distance and time in the planet frame for the explosion event, as well as the distance and time in the spaceship frame.

Next, I would consider the motion of the spaceship and its distance from the planet at the time of the explosion. This can be calculated using the Lorentz transformations and the known distance L in the planet frame.

Finally, I would combine these two calculations to determine the distance from the planet at the time of the explosion in the spaceship frame. This can be done by using the Lorentz transformations to convert the distance and time in the planet frame to the spaceship frame.

Regarding the requirement for the origins of the two coordinate systems to coincide at time t = 0, this is not always necessary. The Lorentz transformations can be generalized to account for different starting times, as long as the relative velocity between the two frames is constant. However, in this specific problem, it may be easier to consider the origins of the two frames to coincide at the time of the explosion, as it simplifies the calculations.

In conclusion, as a scientist, I would approach this problem by breaking it down into smaller parts and using the Lorentz transformations to calculate the relevant distances and times in each frame. This would allow me to determine the distance from the planet at the time of the explosion in the spaceship frame.