How Does Relativity Explain Shorter Travel Times at High Speeds?

[SAT2400]

Relativity(Light years)? PLEASE HELP!

Homework Statement

Imagine an astronaut on a trip to Siris, which is 8 light years from Earth. On arrival at Siris,
the astronaut finds that the trip lasted 6 years. If the trip was made at a constant speed of .8c, how can the 8 light year distance be reconciled with the 6 year duration?

Homework Equations

E=mc^2

The Attempt at a Solution

I don't know how to solve this question...8 light years-> 6 years. 3/.8c??
Please help!

 

[tiny-tim]

Hi SAT2400!

(try using the X² tag just above the Reply box )

Imagine an astronaut on a trip to Siris, which is 8 light years from Earth. On arrival at Siris,
the astronaut finds that the trip lasted 6 years. If the trip was made at a constant speed of .8c, how can the 8 light year distance be reconciled with the 6 year duration?

(e = mc² has nothing to do with it … and i think it's Sirius )

the astronaut's measurement of the time (on his own clock) is 6 years

what is his measurement of the distance?

 

[seagulloftime]

The relevant equations are:

γ=1/root(1-v2/c2)
t'=γt
x=γx'

The 6 years is measured from his frame of reference, you need to measure the distance in the same frame as well.

 

[SAT2400]

hmm, can you explain more easily/??:(

Thanks !

 

[tiny-tim]

Hi SAT2400!

Do you know the Lorentz transformation equations?

(you didn't mention them in your "Relevant equations")

If not, look them up in your book. ​

 

[SAT2400]

root of( 1-v^2/c^2) ...is this right??

hmm...could you please explain more in detail??

Sorry,,but I still don't know how to solve this question...:(

 

[tiny-tim]

root of( 1-v^2/c^2) ...is this right??

Sort-of …

but how are you going to use it?

 

[SAT2400]

I still have no idea...T_T

If I knew how to do this,,I would have not come to this website...

SO...

Please help!

THank you very much!

 

[tiny-tim]

Use x and t for the coordinates in the Earth's frame (so Sirius is at x = 8, for all t).

Use x' and t' for the coordinates in the astronaut's frame.

Start the trip at (0,0) in both frames.

What do you get? ​

 

[Matterwave]

For one such as yourself, who seems to have a very rudimentary understanding of SR, I wouldn't bother with Lorentz transforms. For the question all you need to utilize is Length-contraction.

To the people traveling on the rocket, the length between Earth and Sirius is contracted to: $$L=L_0\sqrt{1-\frac{v^2}{c^2}}$$ The time that they measure is then simply: $$t=L/v$$

Now you will notice that since L is contracted, it is no longer 8 light years but something shorter, and hence t can be shorter than 8 years without the rocket traveling faster than c.