Frame of Reference- arrow traveling through a tube

[momentum2357]

I was just wondering about the following problem. Suppose that you have an arrow placed in a tube. If the arrow travels at a relativistic speed, does there exist a frame of reference such that the arrow is completely in the tube with extra tube at its ends? Does there exist a frame of reference such that the arrow overhangs the tube from both sides, that is, both ends of the arrow are visible on both ends of the tube? Justify your answer uwing mathematics, if possible.

 

[Doc Al]

If the arrow travels at a relativistic speed, does there exist a frame of reference such that the arrow is completely in the tube with extra tube at its ends? Does there exist a frame of reference such that the arrow overhangs the tube from both sides, that is, both ends of the arrow are visible on both ends of the tube?

Yes to both.

Justify your answer uwing mathematics, if possible.

Do it yourself. Hint: Don't forget the relativity of simultaneity.

 

[momentum2357]

What I have is that the frames of reference are the tube and arrow, respectively. I concluded this using the simple length contraction formula L=L_0\sqrt{1-\left(\frac{v}{c}\right)^2}}<L (how can I use LaTeX on these forums?). Correct?

 

[Doc Al]

Yes. If the proper lengths of tube and arrow equal $L_0$, then the length of one as measured in the frame of the other will be Lorentz contracted per that formula. (To use Latex, read https://www.physicsforums.com/showthread.php?t=8997" or click on the $\Sigma$ format command.)

 

[country boy]

Suppose the tube has doors on both ends that can be closed and opened. When the arrow is at rest in the frame of the tube it is longer than the tube, and when it is in motion it is shorter than the tube. The operator who is stationary with the tube quickly and simultaneously closes and reopens the doors when the arrow is completely inside the tube. However, the observer moving with the arrow sees the tip of the arrow enter the tube and before the tip reaches the far end of the tube, the door on that end closes. The door then immediately reopens to allow the tip to continue on its way. Subsequently, when the trailing end of the arrow enters the tube, the door on that end closes behind it (and then reopens). In the frame of the arrow the doors do not close simultaneously, and that allows the arrow to extend beyond the tube.

Food for thought: What happens if the doors are not reopened?