Question

Aizam wish to visit the red giant star Betelgeuse, which is 430 light-years away,
and that he want his 20,000 kg spaceship to move so fast that he ages only 20
years during the round trip. How fast must his spaceship travel relative to the
earth

Answer 1

So, he must go 430 light years in 10 years..?! Which is faster than the speed of light... Is this in a relativity course?

- it would be possible with distance contraction if you were going at a a good fraction of the speed of light but the maths for this eludes me right now...

Answer 2

The main math you need is \[\gamma = \frac{ 1 }{ \sqrt{1-\frac{ v ^{2} }{ c ^{2} }} }\]
There are a couple ways you could set this up. for instance, you could multiply the 10 years (each way) by by gamma to find time an earth observer would say the trip takes, but since you don't know the speed. I think no matter how you set it up you'll have to do it with variables in two places and solve for v.
I would use the rocket frame - assume it is not moving and that the planet is moving toward the rocket at speed "v." Use 10 years as the time for the planet to reach you, but the distance it must travel is d = 430 ly / gamma due to the length contraction of space.
Then use v = d / t (where d is the length contracted distance and t is 10 years) and that should be the velocity.
Of course, gamma has velocity in it (which is unknown), so you'll have to do some algebra to solve for v.

Answer 3

Hmm. You're going to have to ignore acceleration, since that requires general relativity.