Understanding Time Dilation and Age Differences in Twin Paradox Experiments

[Raziel2701]

Homework Statement

Twins, A and B. A goes off in a straight line traveling at .96c for 7 years as measured on his clock, then reverses and returns at half the speed. B remains at home. When they return, what is the difference in ages between A and B?

The Attempt at a Solution

My answer is 19.958 yrs. I got this by calculating time dilation for B in the 7 year leg of the trip of A, giving me 25 years. Then I calculated the distance A traveled in 7 years by multiplying .96c times time. I'm a bit iffy on that step but let me continue.

Having found that distance I again found the time by doing the simple distance over velocity and I get 14 years for A, then doing time dilation for B, I get 15.95 years, so A ages 21 years and B ages 40.95 years, thus their age difference is 19.95 yrs.

My question is, is this right? Do I not have to take into account length contraction to calculate the distance A traveled in 7 years or is it because I'm using a relative velocity that makes this calculation ok?

 

[Raziel2701]

I think I saw the error of my ways, I should have multiplied the 25 years times the .96c to get the distance A traveled as measured by B. then calculate the time by dividing by .48c, this is the time it takes A to travel that distance on its way back. Then I found the time dilation effect on B and to make a long story short I got 24.39 yrs.

 

[collinsmark]

I think I saw the error of my ways, I should have multiplied the 25 years times the .96c to get the distance A traveled as measured by B. then calculate the time by dividing by .48c, this is the time it takes A to travel that distance on its way back. Then I found the time dilation effect on B and to make a long story short I got 24.39 yrs.

That sound's a little better. But you might want to check the significant figures during your calculations. The last two digits (on the right side of the decimal) seem a little off.

Also, your wording sort of worries me just a little. You mentioned "the time dilation effect on B." That's not necessarily wrong at all if you do things carefully. It's just that since A is the one that ends up doing the acceleration (B is the frame of reference that is in an inertial frame continuously throughout the whole process -- not A), I would have personally applied the time dilations to A.

Can you tell me which one ends up being older, A or B?

 

[Raziel2701]

So time dilation occurs to A, his time expands, his clock runs slow with respect to B. I always get this mixed up with length contraction, in the sense that twin B would measure length contraction.

Thanks for pointing that out. Twin B would be the older one. I'll check my sig figs but I wanted more than anything else the conceptual understanding.

Thanks!

 

[collinsmark]

but I wanted more than anything else the conceptual understanding.

I know it can be somewhat confusing. The thing to realize in this problem is that frame A,

• Leaves B,
• Turns around at some point,
• and later arrives to meet B once again.

At each of these events in spacetime A accelerates. A is not in what is called an "intertial" frame for the whole procedure.

B on the other hand, never accelerates. Thus B is in an inertial frame the whole time.