How Do You Calculate Velocity in Relativity Without a Calculator?

[edd0107]

I know that the proper length from a person on Earth to a certain constellation is 9 light years and that the time spent traveling in a spaceship during a one-way trip to the constellation is 12 years. I have been asked to find the velocity and I have been told that I will not need to use a calculator. This has me a little stumped.

My thought is that v=d/t so v=9/12 and the velocity is .75c (c is of course the speed of light). That makes sense if I don't need a calculator. This is the only solution I can come up with.

Any comments, suggestions, or confirmation would be helpful?

 

[Doc Al]

Sounds good to me (assuming that the trip time of 12 years is according to Earth observers).

 

[JesseM]

I know that the proper length from a person on Earth to a certain constellation is 9 light years and that the time spent traveling in a spaceship during a one-way trip to the constellation is 12 years. I have been asked to find the velocity and I have been told that I will not need to use a calculator. This has me a little stumped.

Can you use a pencil and paper, or are you supposed to find a way to solve it that doesn't even require you to solve any equations? If the ship travels at speed v, naturally the time taken in the Earth's frame will be 9/v. But because of time dilation, the time for the ship will be smaller than this be sqrt(1 - v^2/c^2), and if we're using units of light years and years c = 1 so the time according to the ship's clock will by a factor of (9/v)*sqrt(1 - v^2), and that gives us (9/v)*sqrt(1 - v^2) = 12. With a little algebra you can solve this for v without needing to use a calculator (although you have to remember your squares table). I get an answer which is different from 0.75c.

 

[JesseM]

Sounds good to me (assuming that the trip time of 12 years is according to Earth observers).

edd0107 said "the time spent traveling in a spaceship during a one-way trip to the constellation is 12 years", I assumed that to mean that the time was 12 years for the people aboard the ship.

 

[Doc Al]

edd0107 said "the time spent traveling in a spaceship during a one-way trip to the constellation is 12 years", I assumed that to mean that the time was 12 years for the people aboard the ship.

You're right. (D'oh!) And that more interesting problem is also easily solved without a calculator.