Spaceship travels at .83 c, relativity

[harts]

Homework Statement

Suppose our sun is about to explode and we escape in a spaceship toward the star Tau Ceti. When we reach the midpoint of our journey, which takes place at v=.83 C we see our sun explode and, unfortunately, we see Tau Ceti explode as well (we observe the light arriving from each explosion. (there's a part a and b but i understand those parts, so i'll go straight to c and d)

c)In the spaceship frame of reference, how long before we saw the Sun explode did it actually explode? (enter a positive value for times in the past.)

d) In the spaceship frame of reference, how long before we saw Tau Ceta explode did it actually explode?

Homework Equations

Δt = $\gamma$(Δt' - v/c²Δx') } S'-->S

The Attempt at a Solution

c) I have worked out an answer using the equation I gave but I'm not sure if its correct.
I said that S is my frame for the spaceship and S' is my frame for the sun.
Δt = $\gamma$(Δt' - v/c²Δx')
so Δt = 1/(sqrt(1-(.83c)²/c²))*((6 years)+(.83c*6c)/c²)
Δt=19.686 years
Δt' = 6 years
So it actually exploded 13.686 years ago?

for part d), i used the same equation but instead used -v
so Δt = 1/(sqrt√(1-(-.83c)²/c²))*((6 years)+(-.83c*6c)/c²)
= 1.823 years
so it actually happened 1.823-6 = -4.171
4 years into the future? This is the main reason why I'm not believing my answer.

 

[tiny-tim]

welcome to pf!

hi harts! welcome to pf!

Δt=19.686 years
Δt' = 6 years
So it actually exploded 13.686 years ago?

how can you subtract measurements in two different frames?

(and what do you think "∆" means? )

 

[harts]

Thanks for the reply!
OK, I thought I was finding the difference in time between the two frames, which is why I subtracted the change in time from one frame from the change in time in the other frame. Am I misinterpreting the question?

"In the spaceship frame of reference, how long before we saw the Sun explode did it actually explode?"

 

[tiny-tim]

hi harts!

(just got up :zzz:)

Am I misinterpreting the question?

"In the spaceship frame of reference, how long before we saw the Sun explode did it actually explode?"

yup! …

both measurements must be made in the spaceship frame …

the time the sun exploded, and the time the light reached the spaceship

 

[harts]

OK I think I understand it now. I know that the sun actually exploded 6 years ago, but because of relativity I have to use a lorentz transformation to figure out how much time it took for us to see it.

For part c, I use my equation

t'= ((-6 years)-(.83c/c^2)(-6ly))/ sqrt(1-(.83c)^2/c^2)= -1.83

For part d, I use the same equation but use 6 light years for my x value and I got -19.686 years.

I guess it just took me a while to understand those lorentz transformation equations. Thanks tiny tim