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

[Mrbilly]

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 = $\gamma$vL/c^2
time and length dilation
t=$\gamma$t'
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?

 

[cepheid]

Welcome to PF Mrbilly!

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 = $\gamma$vL/c^2
time and length dilation
t=$\gamma$t'
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.

 

[Mrbilly]

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...

 

[cepheid]

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).

 

[Nickie]

I would approach this problem by first considering the time dilation and length contraction effects on the events. From the perspective of the spaceship, the two explosions would appear to occur at different times due to the time dilation effect. This means that the explosion in Aspen would appear to occur first, as the time between the two events would be shorter in the frame of reference of the spaceship.

To calculate the exact time difference between the two explosions, we can use the Lorentz transformations. Since the spaceship is moving at a speed greater than 0.6c, we can use the equation t=\gamma(t' + vx'/c^2) to find the time difference between the two explosions in the frame of reference of the spaceship.

Substituting the values given in the problem, we get:

Δt = \gamma (0.0003s + (0.6c)(150km)/c^2)

= \gamma (0.0003s + 90km/c)

= \gamma (0.0003s + 0.0003s)

= 0.0006s

This means that, in the frame of reference of the spaceship, the explosion in Aspen would occur 0.0006s before the explosion in Denver. This confirms that the first explosion occurs after the second in the frame of reference of the spaceship.

Furthermore, we can also consider the length contraction effect. Since the spaceship is moving towards the explosion in Aspen, the distance between the two events would appear shorter in the frame of reference of the spaceship. This would also contribute to the shorter time difference between the two events.

In conclusion, we can show that in the reference frame of a spaceship moving at a speed greater than 0.6c from Denver to Aspen, the first explosion occurs after the second due to the combination of time dilation and length contraction effects.