Okay, round two. Sorry for the bump and double-post, but it's for great justice, I swear.
Determining the time dilation. This is where I fudged some numbers, specifically to do with the acceleration with the craft. You might want to double-check the math if you're wanting this for an assignment or something, I don't know why but numbers sometimes come back to me in the wrong order. I'll have all formulas down to show how I came up with the numbers to make it easier to check. Without an exact quadratic function for acceleration, I'll make it easy and assume that it instantaneously jumps to 0.071c for two years, then instantly to 0.12c. I'm also using civil time measurements. 60 seconds to a minute, 60 minutes to an hour, 24 hours to a day, 365 days to a year, all exact, so not accounting for leap years or anything else. The formula to determine time dilation is as follows:
Δt=Δt'/Δ1-(v²/c²)
Where Δt is time passed on Earth, Δt' is time passed on the craft, v is the velocity of the craft and c is the speed of light. So...
Accounting for the first two years at the lower speed:
2 Earth years = x craft years/√1-(0.071c²/c²)
2 Earth years · √1-(0.071c²/c²) = x craft years
63072000s · √1-0.071
63072000s · √0.929
63072000s · ~0.9638 = 60791724s on the craft, or ~1.928 years.
Accounting for the next forty-eight years at the higher speed:
48 Earth years = x craft years/√1-(0.12c²/c²)
48 Earth years · √1-(0.12c²/c²) = x craft years
1513728000s · √1-0.12
1513728000s · √0.88
1513728000s · ~0.9381 = 1420002733s on the craft, or ~45.028 years.
50 Earth years = ~46.956 years on the craft.
For the second question, first to determine the speed of the radio transmission. It should be the speed of light, but to make sure...
Einstein's paper on special relativity. Einstein's special relativity theory expresses that the speed of light is absolute regardless of everything. Looks scary, but we only need concern ourselves with
this formula.
V=(v+w)/(1+(vw/c²))
Where V is the observed velocity of the transmission, v is the velocity of the craft, w is the absolute velocity of the transmission, and c is the speed of light. Because w=c, there's no number crunching here, just canceling out variables.
V=(v+c)/(1+(vc/c²))
V=(v+c)/(1+(v/c))
V=(v+c)/(v+c)/c
V=c=w
This leads me to question myself with the Doppler hertz thing, but at least this part is done.
Now, if on day 60
in the craft, a radio transmission from the craft was sent to Earth...
x Earth days = 60 craft days/√1-(0.071c²/c²)
x Earth days = 60 craft days/√1-0.071
x Earth days = 60 craft days/√0.929
x Earth days = 60 craft days/~0.9638
~62.25 Earth days = 60 craft days.
But that's when it was sent. It won't be received until...
60 days = 5184000s
At 0.071c or 21285264.52 m/s
Comes out to 110342811271680 meters.
Transmission travels at c, so transmission will be received 368064 seconds or 4.26 days after sending. That isn't craft days or Earth days, but absolute days, as light travels at the same speed for all observers of all relations (Einstein rocks!!!).
So it will be on Earth day ~66.51 that the message is received.
Earlier today I was coming up with figures that would imply that the craft would be going
back through time which physics states is impossible, so... be sure to double-check the math, that's absolutely important if you're doing this for a class.
*** JohnDoe high-fives himself and takes an aspirin. ***
