In this variant of the "long ladder in the short garage paradox", we will get "out of the box" a little and use an arrow and an openended box to illustrate, as shown in figure 1. The conventional discussion will first be given and then followed by a more 'intuitively accessible' one.
Figure 1
If the arrow is moving lengthwise at a significant portion of the speed of light, then in the rest frame of the box, the arrow is length contracted so that it could, at least for a moment, fit completely into the box, before beginning to exit on the far side.
Figure 2
Light moves at very, very close to 1 nanosecond (ns) per ft, so even without relativity, you can check that the front of the arrow will take 12.5ns to travel the length of the 10ft box. Two clocks, one at each end of the box and synchronized in the reference frame of the box measure this time interval.
However, we are entitled to view the arrow as the rest frame and let the box move towards the arrow at the same speed. Now, the box is length contracted and the arrow cannot, even for a moment, fit into the box.
Figure 3
Again, without using relativity, one can check that the left end of the box, moving at 0.8c, takes 18.75ns to travel the 15ft length of the "stationary" arrow. Two clocks synchronized in the reference frame of the arrow, one at the head and one at the tail, measure this time interval. Note the peculiar order of events  it appears as if events 2 and 3 are chronologically swapped around, compared to what it was in the frame of the box.
A natural question at this point is: "are these differences in the timing of events caused by relativistic time dilation?" The answer is, quite surprisingly, no, not quite! This miniseries will show how the "paradox" it is properly explained by a combination of time dilation and the way clocks are synchronized in inertial frames (which is essentially the same thing as relative simultaneity).
As usual, you can read more on these concepts on the website Relativity 4 Engineers.
Jorrie
PS: "Relativistic length contraction" is a very ugly concept, is it not? We will use a slightly "out of the box" model that does not require contraction, so watch this space!
J

Re: Paradoxes of Relativity Part 3  Ladder Paradox (i)