Astronaut travels to distant planet, finding age?

[theafonis]

Homework Statement

A 26 year-old biologist makes a trip to study alien life forms in a distant planet 10 light-years away. The round trip including a stay of 1 year in the distance planet takes 21.5 years according to the clock on earth. The biologist’s son is 3 year-old when she left. Assume that the planet is not moving with respect to earth, and the speed is the same for both outbound and inbound part of the journey.

What is the difference in age between the mother and son after she returns back to earth?

Homework Equations

T' = T √(1-(v2 /c2))

The Attempt at a Solution

A. 7.0 years B. 16.8 years C. 23.0 years D. 30.0 years E. 37.0 years

The answer is 7 years according to the solution manual. I tried to find the speed of the astronaut, then plugged it into the relevant equation. I'm not sure how to find the speed of astronaut actually. I think I have to solve for the reference frame of the astronaut T, as T' is the frame of earth.

 

[jbriggs444]

I'm not sure how to find the speed of astronaut actually.

Don't over-think it. You know the distance travelled. 10 light years out, 10 light years back. You know the time the journey took. 21.5 years minus the 1 year layover = 20.5 years.

 

[theafonis]

Don't over-think it. ou know the distance travelled. 10 light years out, 10 light years back. You know the time the journey took. 21.5 years minus the 1 year layover = 20.5 years.

(20ly)/(20.5) = 0.97c
Using that, I solved for T to get 4.8 years. About 5 years to the astronaut's age is 31, and the son is 24.5, the difference is 7 then. I think this is correct.

 

[mfb]

I get exactly 4.5 years for the time the astronaut gets older while on the spaceship, while the son ages by 20.5 years => astronaut gets 31.5, son gets 24.5 => 7 difference.

The given options here are very convenient - once you see that the son will age more than the astronaut you are down to two options. The timescale suggests a trip very close to the speed of light, it is possible to see that the second option does not fit even without actual calculations, which just leaves one option.