Complex relativity question (not homework)?

... "two rockets are now 372,000 miles apart (1st is that right?)"

Yes.

..."they have traveled 372,000 miles away from each other in a period of 1 second. (2nd is that right)"

Yes.

..."relative to one another the rockets have moved apart at nearly twice the speed of light. (3rd is that right)"

No.

"Assuming I'm correct that the rockets are now 372,000 miles apart in a period of one 1 second, how have they not traveled away from one another at faster than the speed of light?"

Your right the rockets in a time span of one second are now (because they moved in opposite directions) 372,000miles apart that's almost 2 light seconds separation in 1 second, but each only travelled almost 1 light second. Neither broke the law. c.

PS : to understand where you went wrong - Simply repeat your scenario with smaller speeds / distance, for instance 10 miles for 1 second.

All the best.

The key to your question is exactly that of relativity: An observer can only observe from his or her frame of reference in space and time. Your example seems to assume that there is an absolute reference in space and time. There is not, hence the word: relativity!

An obserer on rocket A or rocket B, will observe the other vehicle to move at the speed of light, never more. A third observer say, on earth, will also observe it that way. Remember that when you say: The rocket A travels at ... (speed), you are refering to a speed relative to something else. What was it?

Here is an example: I fly my super-duper spaceship at nearly the speed of light, relative to you, on earth. I switch on my headlights and see it illuminating objects in front of me nearly at once.

But, on earth, you see my headlights moving very slowly ahead of my spaceship. How is it possible? Ah, the thing is that, what appears to be one second to me, appears to be one year to you! This is called, time dilation.

Another example: Last week, I went flying my super-duper spaceship and when I came back one week later, the Space Agency asked me Space Taxes for one week. I answered: "Sorry but for me, in the spaceship, I was away only for one day and I don't intend to pay for one week! ... By the way, how do you know that it was me who flew away from you and not the earth that flew away from me?"

- "Good try!" they answered, "but we have checked the inertia navigation system in your spaceship and it shows that you accelerated for three days, then decelerated for three days, hence you move from us and not the opposite!"

And here is the answer: In order to experience time dilation, you need to change the speed, acceleratate or decelerate. The interesting thing, here is that, in physics, we can't differentiate between acceleration and gravitation! But I digress ;-)

Oh, one thing more: It was a thought experiement, I don't have a spaceship and, even if I had, at an acceleration of one G, (what is comfortable for the human body) it would take me about 40 years to reach only half the speed of light so ... kids, don't try this at home unless you have plenty of time! ;-)

They are going in different directions. Going in different directions does not mean you are going any faster.