Relativity problem in a text not homework though

[mt8891]

Since it's summer and I'm only taking a chem class I picked up my physics text and started browsing. I found the following problem and I was able to get an answer close to what the book has in the back..I can't really explain why that is so. it's my opinion on the matter that even though I found an answer pretty close it doesn't matter because I don't really know why. anyway, the problem goes something like: Suppose the sun is about to explode. In an effort to escape, we depart traveling in a spacecraft at the velocity .800c and head towards the star Tau Ceti, 12.0 lightyears away. At the midpoint of our journey from Earth we see our sun explode, and at the same time see Tau Ceti explode. In the spacecraft s frame of ref, should we conclude that the two explosions occurred simultaneously. If not which occurred first?

since I've only read the Papers on SR and GR, and labored over the conceptualizations without an adequate mathematical foundation I proceeded as follows:

at first I figured that the sun exploded first. I don't know why but I did. Looked up the answer and it was wrong. This made sense since light travels the same speed regardless of the reference frame. I figured then that the light from the sun must have traveled a shorter distance, since the spacecraft was only traveling at a fraction of the speed of light and the inverse is true for Tau Ceti.

This is where I stop really understanding what's going on and start guessing and checking. I proceeded to multiply the 'distance' traveled, 6 ly the midpoint, by .8 . I figured this was the distance the light from the sun exposion had to travel to reach the space craft. It turned out to be 4.8. For the other side I divided by .8 and got 7.5. When I took the difference (2.7) and multiplied it by 6, I obtain 16.2. The answer in the back is that the explosion of Tau Ceti occurred 16.0 years before the sun. The fact that the answer is to 3 significant figures gives me an indication that I'm incorrect so I really don't care how wrong I am, I'm more concerned about how to obtain the answer.

 

[Mentz114]

At the midpoint of our journey from Earth we see our sun explode, and at the same time see Tau Ceti explode. In the spacecraft s frame of ref, should we conclude that the two explosions occurred simultaneously.

The conclusion is correct - from the spacecraft 's point of view, they did explode simultaneously. The same would be true for any observer equidistant from the stars.

If not which occurred first?

It depends where you are. Simultaneity is relative, there is no absolute before or after for spatially separated events. Some observers would see one star explode before the other.

 

[Naty1]

My thinking is the same as Mentz in post #2...

Your calculation of 6 ly x 0.8 does NOT appear to be the midpoint...the midpoint IS 6 ly, not 4.8ly. Seems like if you are at the midpoint you have traveled 6/0.8 or for 7.5 years... and have 7.5 more years travel remaining...

And if you were at 4.8ly, then you have 7.2 more ly to go [for a total of 12 ly] at a fixed speed, not 7.5.

I don't get the book answer..are you sure you stated the problem as given to you??

 

[mt8891]

I don't get the book answer..are you sure you stated the problem as given to you??

yeah, the problem is how it is in the text.

edit:
It took some trial and error but I've got it.

I took the length contraction for 6ly proper as was measured by the folks on the spaceship.

6 ly * $$\sqrt{1- (.8c)^2 / c^2}$$ = 3.6 ly

after some fridge logic reference frames:

3.6 ly + 6.0 ly = 9.6 ly * $$(6 years/3.6 ly)$$ = 16 years

so yeah, the "actual" distance the light from Tau Ceti had to travel multiplied by the ratio between the interval it would take light (from our sun) to get to the mid point and the actual distance it traveled according to an inertial observer after she took into account the contraction of the length (as observed by the spaceship).

 

[weaselman]

I am not sure where your 9.6 figure came from, or 6.0 for that matter - if you are in the ship's reference frame, the distance to both stars is 3.6 light years, not 6.
The solution to the problem is simple if you consider the ship stationary, while the stars are moving at 0.8c. The Sun is moving away, so the ship sees the light, emitted from it catching up at only 0.2c, and thus it takes it 3.6/(1-0.8)=18 years to reach the ship. The other star is coming toward the ship, so the light is coming on at 1.8c, and takes 3.6/1.8=2 years to reach the ship. So the time interval between the explosions is 18-2=16years.

 

[mt8891]

I am not sure where your 9.6 figure came from, or 6.0 for that matter - if you are in the ship's reference frame, the distance to both stars is 3.6 light years, not 6.
The solution to the problem is simple if you consider the ship stationary, while the stars are moving at 0.8c. The Sun is moving away, so the ship sees the light, emitted from it catching up at only 0.2c, and thus it takes it 3.6/(1-0.8)=18 years to reach the ship. The other star is coming toward the ship, so the light is coming on at 1.8c, and takes 3.6/1.8=2 years to reach the ship. So the time interval between the explosions is 18-2=16years.

I probably saw such a figure but thought 1.8c was just too odd for me.