Solution to Time Dilation Problem

[atarr3]

Hey all, I'm having a bit of trouble with this problem.

Two events occur in an inertial system K at the same time but 4 km apart. What is the time difference measured in a system K' moving between these two events when the distance separation of the events is measured to be 5 km?

I though I could use the length contraction equation to find the speed the frame is traveling and use that in the time dilation equation, but I don't know the proper time. I found the speed of frame K' to be 0.6c with respect to K.

Thanks in advance for any help you guys give.

 

[JesseM]

Do you know how to use the Lorentz transformation? The position of the origin is arbitrary, so you can just say one of the two events occurs at the origin (x=0 and t=0 in system K), and then in the K frame the other must be at x=4 km, t=0 since they are simultaneous. Then you can use the Lorentz transform to get x' and t' for the second event in the K' frame...

 

[atarr3]

Yeah x' is given to us in the problem. It's 5 km. So when I use the equation t' = [t-(vx/c^2)]/0.8, I get t' = 1.25e^-5 s which would be the time difference, but the book says 1.0e^-5.

 

[JesseM]

Yeah x' is given to us in the problem. It's 5 km. So when I use the equation t' = [t-(vx/c^2)]/0.8, I get t' = 1.25e^-5 s which would be the time difference, but the book says 1.0e^-5.

I think you just made a math error somewhere. The math in these problems is always easier if you use units where c=1, so instead of x=4 km, plug in x=(4/299792.458)=1.334e^-5 light seconds, and since t=0 this gives t' = -0.6*(1.334e^-5)/0.8 = -1.0e^-5 seconds

 

[atarr3]

Ah yes I got it now. I had accidentally put in 5 instead of 4. Thank you so much for your help!