How old is Jacob when Maggie returns from her trip to Barnard's Star?

[renegade05]

another special R question... need help

Homework Statement

Maggie, eleven years old, is very jealous of her younger brother Jacob. Jacob, at
ten years younger, is just a baby, and Maggie feels that he gets all the attention at home.
She sneaks onto a spaceship leaving for Barnard’s Star, 5.96 light years away from the solar system. She is hoping that by the time the ship gets back, she will be the baby of the family. Her plan works and when she returns, she is (biologically) a year younger than Jacob. How old is Jacob when Maggie gets back from her trip? Assume that the trip to Barnard’s Star and back happened at a constant speed and that the ship spent very little time at Barnard’s Star itself.

Homework Equations

v=d/t
Tsqrt(1-(v/c)^2)=T'

The Attempt at a Solution

Ok my attempt:
So the problem does not give a velocity however I think I still need to use v=d/t. d=(5.96*2). Thus v=11.92/t
Now I want maggies age(M) when she gets back to be jacobs age(J) - 1.
So:
before:
M = 11
J = 1
After:
M = J -1
J = ?

I am not sure what to do next now! or if this is even right ?

Need some guidance.

 

[tiny-tim]

hi renegade05!

call the speed v

then find the formula for the age increase of both

show us what you get ​

 

[karlo456]

I'm not sure where to go from here, I have a formula for the age increase of Maggie and Jason but I'm stuck on what to do next. I tried solving for v but that proved impossible. Anyway this is what have, if anyone can suggest a next step I would greatly appreciate it!

d=5.96ly
t_J=(2*5.96ly)/v=11.92/v (this is the time increase from Jasons FoR)
t_M=(1/γ)(11.92/v) (this is the time increase from Maggies perspective)

J=1+11.92/v
M=11+(1/γ)(11.92/v)

M=J-1
(substituting)
11+(1/γ)(11.92/v)=11.92/v

Am I on the right track? I can't seem to solve this last equation and neither can WolframAlpha.

 

[vela]

Looks good. You might find it easier to solve for $\beta=v/c$. You have
$$11\text{ yr} + \frac{1}\gamma \frac{11.92\text{ ly}}v = \frac{11.92\text{ ly}}v$$ which, in terms of $\beta$, becomes
$$11\text{ yr} + \frac{1}\gamma \frac{11.92\text{ ly}/c}\beta = \frac{11.92\text{ ly}/c}\beta$$ where $\gamma = \frac{1}{\sqrt{1-(v/c)^2}} = \frac{1}{\sqrt{1-\beta^2}}$. To get rid of the square root, isolate $1/\gamma$ on one side of the equation and then square both sides.

 

[karlo456]

It's such a messy equation, this is what Wolfram says it should be but I keep getting different answers, too many algebra mistakes I guess!

This is what I get:

v=(22*11.92*c^2)/(11.92^2+121c^2)

but it evaluates to a v ≈ 2.17

 

[vela]

You're messing up the units.