Theres this atomi question i'm confused about and was wondering if i could get some help.
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I got to 6.06 easily enough but I don't get that you don't plug it into the time dilation formula which I did and ended up with 8.5 years. I thought it still went in because it doesn't matter about the observers speed just what is being observed which is relativistic. Maybe I'm just tripping
its been a bit since i did this but hopefully this helps:
firstly, relative to earth, the speed of the ship is 0.72c. that means that, according to an observer on earth, that is how fast the ship is travelling. hence, we can just use the classical formula t = d/v to determine the time taken, according to an observer on earth, for the ship to reach the distant planet. we don't need to use time dilation here, because the person on earth is measuring the proper time according to themselves.
proper time is said to be measured by the person "at rest to a clock", or essentially, someone in the same reference frame as the clock being used to measure time (there is no absolute clock). here, we assume the perspective of the person on earth, as we are given information from their perspective - the length they measure to alpha centuri, for example. again, quite easily, the proper time could have been measured by the person on the spacecraft, and then according to the person on the spacecraft, the person on earth will be the one experiencing time dilation, because there is no absolute reference frame. this is perfectly calculatable, although you might also have to account for length contraction because the length to alpha centuri would appear different to the person on the spaceship, but the question doesn't allow for you to consider this perspective as far as i can see. either way, here we are forced to think from the perspective of the person on earth, which means that they believe that they are measuring the proper time, which means they believe that they aren't experiencing time dilation; so you don't need to use the time dilation formula here.
now for the second part, according to the people in the spaceship, a clock on earth is moving slower, so indeed the person on earth is going to be experience time dilation according to the observer on the spaceship; thus we have to use the time dilation formula. now, the proper time is measured from the perspective of the person in the spaceship, so the time calculated in the previous part is the dilated time, according to those in the spaceship, as they see the clocks on earth to be slowed down. thus, the "t_0" in the answers is really "proper time, from the reference frame of the spaceship; or the dilated time experienced by those on the spaceship, from the perspective of the person on earth", and "t" is really "the dilated time experienced on earth, from the reference frame of the spaceship; or the proper time, from the reference frame of earth".
