Heide boards a spaceship and travels away from Earth at a constant velocity of 0.45c toward Betelgeuse (a red giant star in the constellation Orion). One year later on Earth clocks, Heide’s twin, Hans, boards a second spaceship and follows her at a constant velocity of 0.95c in the same direction.
a) When Hans catches up to Heide, what will be the difference in their ages?
b) Which twin will be older?
Hint given: Work primarily in the Earth's frame, and compute the a) the headstart, in Earth's frame Heidi has before Hans leaves, b) the time in Earth's frame it takes for Hans to catch Heidi. Then apply time dilation carefully or Lorentz transformations to compute how much time passes in each of their frames, keeping in mind that Hans is in the Earth's rest frame for one year.
γ= 1 / √(1-v2/c2)
The Attempt at a Solution
For part a) Using the hint I got that Heide's headstart is .45 cy (light years). After the one year period, Hans is moving at .5c relative to Heide, in Earth's frame. Moving at this speed, again in Earth's frame, it would take him 0.9 years to catch up to Heide. Now I calculated gamma for Hans to be 1.1547, and I am just not sure what to do next. I think that Heide will measure the proper time because now she is in Earth's rest frame as Hans moves towards her. So is this simple enough that I can just plug numbers into the first equation?