Two explosions occur, spaceship is flying overhead at 0.6C, which occurs first?

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Homework Help Overview

The problem involves two explosions occurring at different times and locations, with a spaceship traveling at a significant fraction of the speed of light (0.6c) between them. The task is to analyze the timing of these explosions from the spaceship's reference frame using Lorentz transformations and concepts of simultaneity in special relativity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of Lorentz transformations to determine the timing of explosions from different reference frames. There are attempts to calculate dilated time and length, with some questioning the correctness of their coordinate assignments and transformations.

Discussion Status

Some participants have provided guidance on the correct application of Lorentz transformations, while others express confusion regarding their calculations and the implications of their results. Multiple interpretations of the timing and coordinates are being explored, indicating an ongoing investigation into the problem.

Contextual Notes

Participants note the importance of correctly identifying the reference frames and the transformations needed to analyze the events. There is mention of the need to clarify the direction of transformations between the unprimed (Earth) and primed (ship) coordinate systems.

Mrbilly
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Homework Statement


At 11h 0m 0.0000s AM a boiler explodes in the basement of the Denver Science Museum. At 11h 0m 0.0003s, a similar boiler explodes in the basement of a ski lodge in Aspen at a distance of 150 km from the first explosion. Show that in the reference frame of a spaceship moving at a speed greater than v=0.6c from Denver to Aspen, the first explosion occurs after the second.


Homework Equations


Lorentz Transforms
x=\gamma(x' + vt')
y=y'
z=z'
t=\gamma(t' + vx'/c^2)

Simultaneity
Δt = \gammavL/c^2
time and length dilation
t=\gammat'
L=L'/\gamma


The Attempt at a Solution


I first state that from the ship's perspective, the two boilers are moving, the one in denver away from the ship at 0.6c, the one in aspen towards the ship at 0.6c. I can find that, due to time dilation, the actual time between explosions as viewed from the ship would be 0.00024s, but don't know where to go from there. Also I used the simultaneity equation using Δt=0.0003s to find the distance between the two blasts, which was just length dilation in the end so i did more work for nothing. I am stumped now though. I have both dilated time and length of travel, but how can i show that the explosion in aspen happens first?
 
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Welcome to PF Mrbilly!

Mrbilly said:

Homework Statement


At 11h 0m 0.0000s AM a boiler explodes in the basement of the Denver Science Museum. At 11h 0m 0.0003s, a similar boiler explodes in the basement of a ski lodge in Aspen at a distance of 150 km from the first explosion. Show that in the reference frame of a spaceship moving at a speed greater than v=0.6c from Denver to Aspen, the first explosion occurs after the second.

Homework Equations


Lorentz Transforms
x=\gamma(x' + vt')
y=y'
z=z'
t=\gamma(t' + vx'/c^2)

Simultaneity
Δt = \gammavL/c^2
time and length dilation
t=\gammat'
L=L'/\gamma

The Attempt at a Solution


I first state that from the ship's perspective, the two boilers are moving, the one in denver away from the ship at 0.6c, the one in aspen towards the ship at 0.6c. I can find that, due to time dilation, the actual time between explosions as viewed from the ship would be 0.00024s, but don't know where to go from there. Also I used the simultaneity equation using Δt=0.0003s to find the distance between the two blasts, which was just length dilation in the end so i did more work for nothing. I am stumped now though. I have both dilated time and length of travel, but how can i show that the explosion in aspen happens first?

All you need to know to solve this problem is that the spacetime coordinates of events in two different coordinate systems (reference frames) are related to each other by the Lorentz transformation. You have x1 and t1, which are the location and time of the first explosion in the unprimed (Earth) reference frame. You also have x2 and t2, which are the location and time of the second explosion in the Earth frame. All you have to do is apply the Lorentz transformation to each pair of coordinates (x,t), to find x1ʹ, t1ʹ, x2ʹ and t2ʹ, which are the spacetime coordinates of the two explosions in the primed (ship) reference frame. You should find that t1ʹ > t2ʹ.

Note: this belongs in the Introductory Physics subforum, since the Advanced Physics one is for upper-year undergraduate and graduate-level physics homework only. Thread moved.
 
Thanks for moving the thread, sorry about that.

I did all the math, but i still come up with the time for the denver explosion at 0 seconds and the aspen explosion at somehow a longer time of 0.000375s. Am I wrong in saying that the (x,t) coordinates of the denver boiler are (0,0) and aspen is (150,0.0003)? just because with the (0,0) coordinates, everything will just go back to being 0...
 
Mrbilly said:
I did all the math, but i still come up with the time for the denver explosion at 0 seconds and the aspen explosion at somehow a longer time of 0.000375s. Am I wrong in saying that the (x,t) coordinates of the denver boiler are (0,0) and aspen is (150,0.0003)? just because with the (0,0) coordinates, everything will just go back to being 0...

No you're not wrong. Those two sets of coordinates are correct. Remember that you're going from the unprimed (Earth) coordinate system to the primed (ship) coordinate system, so the transformation equations are:


xʹ = γ(x - vt)
tʹ = γ(t - vx/c2)
​

What you wrote down in your first post was the reverse transformation from this (going from primed to unprimed).
 

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