Physics "Space Travel"
posted by Chelly .
Suppose you want to go to the immense red giant Betelgeuse, which is about 500 light years away. (A light year is the distance that light travels in one year.) You plan to travel at constant speed in a 1000 ig rocket ship.
A: If the rocket ship's speed is 0.500 c, calculate the time for the trip, as measured by people on earth.
B: If the rocket ship's speed is 0.500 c, calculate the time for the trip, as measured by astronauts in the rocket ship.
I'm quite confused about which equation to use for which part... Any help is immensily appreciated!

You can solve all such problems using the fact that a distance in SpaceTime between two points is defined as:
S^2 = (distance in time)^2  (distance in space)^2
The distance in time is defined as the difference in the time coordinate times c (the speed of light).
Because S is a distance betwen two points it does not matter in which reference frame you evaluate it.
Let's compute S^2 between the two points defined as:
1) Spaceship leaves Earth
2) Spaceship arrives at Betelgeuse
According to an observer at Earth, the travel time will be 1000 years. The spatial distance is 500 lightyears. This means that:
S^2 = 1000^2  500^2 = 7.5*10^5 year^2
(I've put speed of light equal to 1 and I'm thus measuring spatial distances in years. I.e. 1 lightyear = 1 year in these units)
Now, we can also evaluate S^2 in the reference frame of the observer. If we do that, we should, of course, find the same result. However, we don't know the time difference between the two points. We do know that the difference in spatial coordinate is now zero.
If we call the travel time in the reference frame of the observer T, then we have:
S^2 = T^2 >
T = sqrt[7.5*10^5] years =
866 years. 
When I use the above I get correct answers for 2 different values, but when the given is .9999c, I don't get a correct answer... Why??
There is also a C Part:
If the rocket ship's speed is 0.9999c, calculate the energy needed in joules.
How do I convert to this??