Calculating Time Dilation at Half the Speed of Light

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Homework Help Overview

The problem involves calculating the speed at which a clock ticks at half the rate of an identical clock at rest, specifically in the context of special relativity. The subject area is time dilation and the relevant equations are related to the Lorentz factor.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the appropriate formula for time dilation and the interpretation of the variables involved. There is an exploration of the ratio of time intervals as observed from different frames of reference.

Discussion Status

Participants have engaged in clarifying the problem setup and have begun manipulating the relevant equation. Some have expressed uncertainty about the next steps in isolating the variable of interest. There is a mix of interpretations regarding the phrasing of the problem and the implications of the moving clock's perspective.

Contextual Notes

There is a mention of a missing variable in the original post, and some participants are reflecting on the nuances of the problem's wording. The discussion is ongoing with no explicit consensus reached yet.

Naoki
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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!
 
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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' ?
 
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.
 
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.
 
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.
 
Hah, didn't even catch that. I'm currently taking this course. Very interesting material!
 
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?
 
looks like I might have got it the answer is .9c. Thanks!
 

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