Calculating the Speed of a Starship Using Time Dilation

[Xaspire88]

A starship Voyages to a distant planet 10 ly away. The explorers stay 1 yr, return at the same speed, and arrive back on Earth 26 yr after they left. Assume that the time needed to accelerate and decelerate is negligible.
a) What is the speed of the starship?
b) How much time has elapsed on the astronauts' chronometers?

At first this seemed very straight forward and I thought i would have no problems with it.
I started off using the equation

$$\Delta t = \frac{\Delta t^1}{\sqrt{1 - v^2/c^2}}$$

The $$\Delta t$$ for this problem should be 25 years since that is the amount of time spent traveling, or at least it would seem to me you would not count the 1 yr they spent on the other planet. But once I arrived at this point i was still missing one variable necessary to complete the problem. One being the speed of the starship(which needed to be found), and the other was the $$\Delta t^1$$ (which also needed to be found). I can not find out how to solve this problem without one of those variables. Help please

 

[Doc Al]

You have the distance the ship traveled and the time it took to get there. Use that data to compute the speed.

 

[Xaspire88]

so i wouldn't use that equation then for part a. I would just take

20 ly/25 yrs?
(20*3*10^8)/(25)= 2.4 * 10^8

2.4/3= 0.80
v= 0.80c

 

[Doc Al]

Right.

 

[Xaspire88]

hmm it seems odd that the time on Earth would give you a correct value for the velocity considering that the distances would change when approaching such great speeds.

 

[Doc Al]

hmm it seems odd that the time on Earth would give you a correct value for the velocity considering that the distances would change when approaching such great speeds.

As long as you stick to measurements made in the same frame, finding speed is as simple as distance/time. Both the distance and time given in this problem are as measured by Earth observers.

 

[Xaspire88]

Oh right because its 10ly away from Earth ... Ok that makes sense. Thank you.

 

[Xaspire88]

So then i would take this velocity and put it into my original equation and solve for deltaTprime... When i do this i get a time value of 15 years, using 25 years as the deltaT. Then I would add one year to that for the time that was spent on the other planet? So their chronometers would read 16 years?

 

[Doc Al]

Yep. That's what I would say.

 

[Xaspire88]

So even though a photon of light would take ten years to get there and ten years to get back.. a total of 20 years. I guess it just seems weird to me that, unless i am thinking about this the wrong way, it would take longer for a beam of light traveling at the speed of light than this spaceship.

 

[Doc Al]

So even though a photon of light would take ten years to get there and ten years to get back.. a total of 20 years. I guess it just seems weird to me that, unless i am thinking about this the wrong way, it would take longer for a beam of light traveling at the speed of light than this spaceship.

The proper comparision is: While the beam of light takes 20 years for a round trip, the spaceship takes 25 years.

 

[Xaspire88]

Because relative to itself a beam of light would travel there and back in a shorter amount of time than the spaceship. I guess i was thinking of the beam of light relative to time on Earth rather than to itself.