How Far Into Earth's Future Do the Planet of the Apes Astronauts Travel?

[DeadxBunny]

Homework Statement

The premise of the Planet of the Apes movies and book is that hibernating astronauts travel far into Earth's future, to a time when human civilization has been replaced by an ape civilization. Considering only special relativity, determine how far into Earth's future the astronauts would travel if they slept for 105 years while traveling relative to Earth with a speed of 0.9900c, first outward from Earth and back again.

Homework Equations

I think delta t = Y*delta t0
where Y = 1/[1-(v/c)^2]^1/2

The Attempt at a Solution

I used the above equations (delta t0 = 105 y, v = 0.99c) and got 744.3253. Then I multiplied this by 2 and got 1488.651 y which is the wrong answer.

Thanks for your help!

 

[Tomsk]

You have to go there and back in 105 years, so you go out for 105/2 years and back for 105/2 years. I've had too much fosters to figure out the real answer, I just hope that helps... hmm its 3.20am here. nice. Wait, why did you x by 2? It should probably just be 744 years. But maybe you should take into acount the the x displacement...

 

[Daverz]

It helps to draw a spacetime diagram. The paths taken by the Earth and by the astronauts will form a triangle. Then use the formula for proper time:

\Delta \tau^2 = \Delta t^2 - \Delta x^2

 

[Max Eilerson]

Won't the length contraction cancel out?

 

[DeadxBunny]

Did I misunderstand the question? Is the spaceship going out and then back to Earth all in 105 years? I thought the question meant that the ship went out for 105 years and the return trip also took 105 years (which is why I multiplied by 2).

You have to go there and back in 105 years, so you go out for 105/2 years and back for 105/2 years. I've had too much fosters to figure out the real answer, I just hope that helps... hmm its 3.20am here. nice. Wait, why did you x by 2? It should probably just be 744 years. But maybe you should take into acount the the x displacement...

 

[Daverz]

Did I misunderstand the question? Is the spaceship going out and then back to Earth all in 105 years? I thought the question meant that the ship went out for 105 years and the return trip also took 105 years (which is why I multiplied by 2).

Well, remember the story for Planet of the Apes. They are asleep the whole time, while the ship gets turned around (which they don't realize) and comes back to earth. So 105 years should be the total trip time, or 52.5 years for the trip out and 52.5 years for the trip back if you assume symmetry.

 

[DeadxBunny]

I never saw the Planet of the Apes lol. It probably would've helped me out with this problem! Thanks for your help, guys! The answer they're looking for is 744 instead of 1488!

Well, remember the story for Planet of the Apes. They are asleep the whole time, while the ship gets turned around (which they don't realize) and comes back to earth. So 105 years should be the total trip time, or 52.5 years for the trip out and 52.5 years for the trip back if you assume symmetry.