1. The problem statement, all variables and given/known data An electron travaels in an accelerator tube at a speed of 0.997 c relative to Earth. In the frame of reference of the electron, the length of the tube is 1.20 m. What is the length of the tube relative to earth? 2. Relevant equations L = Lo / γ Textbook answer: 15.5 m 3. The attempt at a solution I know it is a simple plug-in problem but when I plug in the numbers, they don't work out and I'm guessing that I'm assigning the numbers wrong. For this problem, would 1.20m be L or Lo? I'd assume it to be Lo since that's the proper length in terms of the electron's reference frame. However, the numbers do not work out then - indicating that L = 1.20 m. This does not make sense to me since 1.20 m is the length measured from the electron's frame, is it not?
Yes, 1.2m is the length as measured from the electron's frame. In your formula above, L_{0} is the proper length of the tube as measured in the frame we are working in, and L is the length as observed by the moving observer. So, suppose we are in the frame at rest relative to the earth. Then, L_{0} is the proper length relative to the earth, L is the length which the electron sees, and v (in the factor gamma) is the relative velocity between the earth and the electron.
Hmm. So if 1.20m is from the electron's frame, wouldn't that mean it is assigned to L0? But, since this is the proper length we're dealing with where we have to measure the tube at the same time, it has to be from the earth's frame .. doesn't it? Sorry, I guess I don't understand the assignation/ the nature of the variables. L = L0 / γ Either: L = (1.20m)√(1 - 0.997²) L = 0.093 m Or: 1.20m = (L0)√(1 - 0.997²) L0 = 15.5 m (textbook answer)
The 0 in L0 stands for the proper value, the value in its own restframe. Clearly from the electron's rest frame the tube is not at rest and thus it cannot be assigned L0.