Time Dilation Problem; Find the proper time, ts.

[Ammar SH]

Homework Statement

Alpha Centauri, the closest star to Earth, is 4.3x10^6 m away. How long would it take a spaceship to reach the star if it were traveling at 0.999c?

Homework Equations

I did get the answer as 0.2a...and the textbook also said it would take about 2 months,
I do not understand what the a represents/stands for and how the textbook got 2 months from 2a.
Also, I do not why I had to find tm and then to?

The Attempt at a Solution

ts = d/v = 4.3 x 10^16/0.999c = 1.43x10^8 s

tm = ts/√(1-v^2/c^2
= 1.43x10^8/√(1-(0.999c)^2/c^2
= 3.19 x 10^9 s
tm = 4.5a

THE ANSWER KEy gets the same answer till the part above but then the answer key starts doing more calculations afterwards(below) which I do not get

ts = tm√1-v^2/c^2

= 4.5a√1-(0.999c)^2/c^2
ts = 0.2a

the textbook gave 0.2a but also said it will take two months to reach the star and that is what I do not understand.


Thanks for your help.

 

[Disconnected]

Quick question, could you define all of your variables? a, tm, ts to etc.

Also, is that really the full question? It doesn't state anything about reference frames.

 

[Ammar SH]

yes that is the whole question and you are right it does not tell you the frames...Im guessing you are suppose to figure out the frames yourself..


d from Earth to star= 4.3x10^16 m
v= 0.999c=3x10^8x0.999=2.9997X10^8

I used the distance time formula to find the ts(well I think the answer I found is ts)
ts = 1.43x10^8 s

In case if you do not know: ts is the proper time, the time interval between two events measured by an observer who sees the events occur at one position
and tm is the time interval for an observer moving with a speed v relative to the sequence of events.


And that is what confuses me... I found the right answer but i don't understand how and why

tm=3.19x10^9s is also equal to 4.5a

i mean what is a??! I am going crazy...

Uptill 4.5a I got but then the answer key does the next steps which makes no sense and
says the final answer is 0.2a which is two months.

how is 0.2a = two months?? what does a stand for and what# does it equal?

 

[Disconnected]

"= 3.19 x 10^9 s
tm = 4.5a"

Super confused about what happened here.

Am I missing some unit of time?

Btw, you should probably try to put units into your solutions throughout.

I don't think you found ts, I think you found the time in alpha centuri's reference frame. You can then use time dilation to find out how long it took in the reference frame of the ship. Or you could use length contraction to find the distance in the reference frame of the ship and solve it that way.

 

[Ammar SH]

yea sure I will do that but the book literally states in answers(unless they are wrong)

4.5a=3.19x10^9 secondsi will try one more time

thanks for your help I think I will speak to teacher about this question, maybe something is wrong with it