Calculating Time Dilation at Half the Speed of Light

[Naoki]

Homework Statement

At what speed relative to a laboratory does a clock tick at half the rate of an identical clock at rest in the laboratory. Give your answer as a fraction of c (speed of light)

Homework Equations

Δt=Δt'/√1-^2

The Attempt at a Solution

My professor assigned this question to me to go over in class on our next meeting. We did not go over how to go about solving this. I have no idea where to start or if this is even the right formula to begin with. Please coach me through, Thank you!

 

[ElijahRockers]

Δt is the time interval of the moving clock as observed by someone in the lab. Δt'
is the interval of the moving clock as observed by someone moving with the clock (as in, the same inertial frame of reference as the clock)

what is the ratio Δt/Δt' ?

 

[Naoki]

If I am not mistaken, the ratio as stated in the original question is what you are asking for. That, I think would be 1/2 where Δt=1 and Δt'=2.

Also my original post was missing a variable, which I'm sure you are aware of. It should read

Δt=Δt'/√1-β^2

Sorry for the confusion.

 

[ElijahRockers]

Right.

So using that ratio, move things around and you get

1/2 = √(1-(v/c)^2)

Since C is a constant, there is only one variable left, v. it is the speed of the moving reference frame with respect to the stationary frame, and the answer you are looking for.

 

[phinds]

Just as a matter of nitpickery, it should say at what speed does the moving clock APPEAR to tick at half the speed. As far as the moving clock is concerned, it is the earthbound clock that is slow.

 

[ElijahRockers]

Hah, didn't even catch that. I'm currently taking this course. Very interesting material!

 

[Naoki]

Thank you for that. Now I guess the rest is math in isolating the only variable left, but its not happening for me. Do you mind helping me a bit further?

 

[Naoki]

looks like I might have got it the answer is .9c. Thanks!