Relativity Question (Twin Paradox)

[b2386]

Hi all,

Most of you are probably familiar with the twin paradox of relativity, which is the basis of this problem. I think I understand it fairly well but am having trouble with one specific detail. Here is the setup:

Problem
We have two planets, the Earth and canopus, that are separated by a distance of 99 light years in the Earth frame. A rocket (R1) traveling from the Earth to canopus travels at such a speed that it will arrive at canopus in 101 years. Upon arriving at canopus, the occupant in the rocket will instantaneously jump from the first rocket to a second rocket (R2) traveling the same speed but now pointed towards the earth.

For the earth, there is a string a clocks that infinitely stretch out towards canopus (the positive direction) and away from the direction of canopus (the negative direction). These clocks are synchronized to the clock on earth. A similar set of clocks are attached to the rockets and are synchronized to the clock on the rocket.


There are six events that occur in the problem.

Event 1: R1 leaves earth
Event 2: R1 reaches canopus
Event 3: Rocket station S1 passes earth
(S1 is the station that reaches Earth at the same time that R1 reaches canopus in the R1 frame)
Event 4: R2 leaves canopus
Event 5: Rocket station S2 passes earth
(S2 is the station that reaches Earth at the same time that R2 leaves canopus in the R2 frame)


There are six clocks

Clock 1: on earth
Clock 2: on canopus
Clock 3: on R1
Clock 4: on S1
Clock 5: on R2
Clock 6: on S2


I am told to find the times of all six clocks at each of the six events in the Earth frame.


Attempted Solution
Well, I found the times of all the clocks of, except clock 4 at event 1 and clock 6 at event 6. If I can find out how to obtain the time from clock 4, I can probably find the time for clock 6 as well.

I started by using the equation $$t^2_E - x^2_E = t^2_R - x^2_R$$ where "E" stands for Earth and "R" stands for Rocket. I am attempting to solve for $$t_E$$. $$t_R$$ is equal to 0 since this is the reference event for the problem.

But before I could substitute numbers to this equation, I had to find the value of $$x_R$$ at event 2 and $$x_E$$ at event 3.

For $$x_R$$, I used the value found for $$t_R$$ at event 2.

$$t^2_R = t^2_E - x^2_E$$

$$t^2_R = 101^2 - 99^2$$

$$t^2_R = 400$$

$$t_R = 20$$

Multiply this distance times the rocket speed (which is 99/101 = .98) to give us 19.6 years in distance between the rocket at canopus and the earth. It is also the distance between the S1 clock in event 3 as it passes

the earth. Back to our main equation, $$t^2_E - x^2_E = t^2_R - x^2_R$$, we now have the value of $$x_R$$ (19.6)

For $$x_E$$, I used the value found for $$t_E$$ at event 3.

$$t^2_E = t^2_R - x^2_R$$

$$t^2_E = 20^2 - 19.6^2$$

$$t^2_E = 15.84$$

$$t_R = 3.98$$


Finally substituting all of our values into $$t^2_E - x^2_E = t^2_R - x^2_R$$, I have $$t^2_E - 3.98^2 = 0 - 19.6^2$$. However, I end up getting the squareroot of a negative number. Can someone please show me where things went wrong?

 

[b2386]

Did I not include enough info?

 

[b2386]

Please help! I have to turn this in tomorrow.

 

[b2386]

For a moment, forget about what I posted above and let's see if I can make this simpler.

A rocket is traveling from Earth to Canopus, which is 99 light years away. The rocket reaches canopus in 101 years (so v = 99/101 = .98). There is a clock attached to a rope that is 3.88 light years long (in the Earth frame) and this rope is attaced the the back of the rocket (basically, there is a clock travling the same velocity as the rocket but is 3.88 light years behind the rocket). When the rocket reaches Canopus, what does this clock read in the Earth frame?