When to apply Lorentz transformations and laws of time dilations and length contractions: explanations

Why I can not to use the formulas of time expansion and length contraction and I must applicate, necessary, Lorentz transformations?

The time dilation and length contraction formulas are derived from the Lorentz transform using some specific assumptions. For time dilation the assumption is that the clock is at rest in one of the frames. For length contraction the assumptions are that the endpoints are at rest in one of the frames and that the length is constant.

In this problem you cannot use the length contraction or time dilation formulas because the assumptions are not met. The particle is not at rest in either the lab or rocket frames.

As an aside, I recommend that new students of relativity not use the time dilation and length contraction formulas at all. They are too easy to misapply as in this problem. Just use the Lorentz transform, it will automatically simplify when appropriate, and you will avoid misapplying them when they are not appropriate.

IMO newcomers (and teachers) in SR should avoid both time dilation and length contraction. I add Lorentz transformations too. Too often they are applied mechanically, without understanding how and why. The most basic instrument in SR is invariance of spacetime interval. It's enough to solve all elementary problems.

So I welcome @Aretino 's comment. There is however a weak point. In Aretino's solution use is made of RTV - relativistic transformation of velocities (I prefer "transformation" to "addition" or "composition" but I can't dwell on that here). Now RTV formula is usually established through Lorentz', so it looks that Aretino's solution couldn't avoid them.

That's not true since RTV can be proved using only invariance of interval (admittedly in a longer way). I believe this proof isn't widely known, although I'm pretty sure that there are books containing it. I'm afraid that reporting it here could be deemed as OT.