Derivation of the length-contraction formula i saw the derivation of the length contraction and it goes like this:

what i dont understand is why does interval between the photon striking and coming back have to be compared? why can't I simply compare the time in going from one point to the other plate and derive the correct formula?
 A: You're relying on the time dilation formula.  That formula is valid for a point that is at rest in the observed system (the rocket).  The formula for the time between two events at different locations in both systems (such as the light and the mirror positions) is more complicated.  This ``complication'' ensures that you get the same result for the length contraction as in the original calculation.
A: "why can't I simply compare the time in going from one point to the other plate and derive the correct formula?"
You most certainly can derive the formula that way. Here is an example:
Derivation of Einstein's Time Dilation Equation
A: In order to measure length contraction, the observer has to see it. This requires that light be emitted from the light bulb, travel to the end of the stick, and travel back to the observer. It is important that the light travel the entirety of the stick: length contraction is here in part caused by the lag between the different infinitesimal elements of the stick.  By this I mean the following:
Consider 2 cases: i) the light is emitted at the left end of the stick and ii) the light is emitted at the right end of the stick. Then in i) the light requires time $L/c$ to reach the right end of the stick and then requires $L/c$ to reach the left end of the stick. Case ii) is similar but with right and left ends switching places. Hence, if the observer is at either end of the stick, they must wait a total time of $2L/c$ for the light to travel from the light source to the other end of the stick and back. The "lag" comes in because it takes a slightly different amount of time for the light to hit and return from one element of the stick than it takes for the light to hit and return from the next element. This means that while the observer will record a smaller time for nearer elements, they must wait for the full $2L/c$ for the furthest end of the stick.
Basically, I think the answer is that the 2 comes in when considering how the experiment would actually be carried out.
