Solving the Twin Paradox: Age Difference Between Bill and Ben

[iaberrant]

Homework Statement

Two twins, Bill and Ben are 30.0 years old and they leave Earth for a distant planet 12.0 light years away. The twins depart at the same time on Earth, and travel in different space ships. Bill travels at 0.93c, while Ben travels at 0.70c. What is the difference between their ages when Ben arrives on the new planet ?

The Attempt at a Solution

I am new to Special Relativity and therefore do not have a great knowledge about the topic however I think this question is related to the twin paradox ? Therefore it would also have to do with time dilation which follows the formula t'= t / √ 1 - v^2/c^2 .

I don't know how to start solving this problem and therefore any help and ideas would be greatly appreciated.

Thankyou in adavance !~

 

[Redsummers]

This problem is not exactly the twin paradox. In the formal twin paradox the spaceship was traveling with acceleration = g. And so, things get slightly more complicated.

In your case, however, using the time dilation formula is more than enough. A good way to start is to find out how much time would it take for both spaceships to travel such a distance with their respective speeds, note that you would get the time in the stationary coordinates.

 

[iaberrant]

So what do you mean by stationary points? does that mean the distance to the plant and the speed ?
so if the planet it 12 light years away then
Distance = 12 x 10^16m
= 1.2 x 10^17 m

and then taking Bill's speed of 0.93c, does it need to be converted to m/s ?
so : 0.93 x the speed of light = 278 806 986 m / s

then these two values could be substituted into the formula time = distance/ speed ?

time= 1.2 x 10^17 /278 806 986
=430405284 seconds

Then using this value substitute it into the time dilation formula as the variable t ?

t'= 430405284 / √ 1 - 0.93c^2
t'=1170981193 seconds
= 37.13 years ?
So Bill would have aged 37.13 years ?

If this method is right, i would then repeat it for Ben and substract the two ages to find the difference between the ages?

Is this correct?

 

[Redsummers]

That's correct!
That would be a valid result, but let me point you out something about the process. As you can see here :

So what do you mean by stationary points? does that mean the distance to the plant and the speed ?
so if the planet it 12 light years away then
Distance = 12 x 10^16m
= 1.2 x 10^17 m

and then taking Bill's speed of 0.93c, does it need to be converted to m/s ?
so : 0.93 x the speed of light = 278 806 986 m / s

then these two values could be substituted into the formula time = distance/ speed ?

time= 1.2 x 10^17 /278 806 986
=430405284 seconds

There is a big loss of generality by multiplying the lightyears to get the distance in meters. What you could do some other time is, instead of that, divide directly the lightyears by the coefficient of 'c'.

Briefly speaking: 12/0.93 = '12.903.. years'. And that could be your time.

If you now compare the results between 12.903 years and your '430405284 seconds', you will see that there's a considerable error. But hey! your way is also valid ;) I'm just saying it, so that you can do it faster w.l.o.g.

After that, you can apply the '12.903.. years' directly to the time dilation formula, and your result will automatically be in years. Hope that helps!