  We study the decay law for a moving unstable particle. The usual
time-dilatation formula states that the decay width for an unstable state
moving with a momentum $p$ and mass $M$ is $\tilde{\Gamma}_{p}=\Gamma
M/\sqrt{p^{2}+M^{2}}$ with $\Gamma$ being the decay width in the rest frame. In
agreement with previous studies, we show that in the context of QM as well as
QFT this equation is \textit{not} correct provided that the quantum measurement
is performed in a reference frame in which the unstable particle has momentum
$p$ (note, a momentum eigenstate is \textit{not} a velocity eigenstate in QM).
We then give, to our knowledge for the first time, an analytic expression of an
improved formula and we show that the deviation from $\tilde{\Gamma}_{p}$ has a
maximum for $p/M=\sqrt{2/3},$ but is typically \textit{very} small. Then, the
result can be easily generalized to a momentum wave packet and also to an
arbitrary initial state. Here, we give a very general expression of the
non-decay probability. As a next step, we show that care is needed when one
makes a boost of an unstable state with zero momentum/velocity: namely, the
resulting state has zero overlap with the elements of the basis of unstable
states (it is already decayed!). However, when considering a spread in
velocity, one finds again that $\tilde{\Gamma }_{p}$ is typically a very good
approximation. The study of a velocity wave-packet represents an interesting
subject which constitutes one of the main outcomes of the present
manuscript.\textbf{ }In the end, it should be stressed that there is no
whatsoever breaking of special relativity, but as usual in QM, one should
specify which kind of measurement on which kind of state is performed.
