  The time evolution of protons and $^3$He fragments from Au+Au/Pb+Pb reactions
at 0.25, 2, and 20 GeV$/$nucleon is investigated with the potential version of
the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) model combined with
the traditional coalescence afterburner. In the coalescence process, the
relative distance $R_0$ and relative momentum $P_0$ are surveyed in the range
of 3-4 fm and 0.25-0.35 GeV$/$c, respectively. For both clusters, a strong
reversed correlation between $R_0$ and $P_0$ is seen and it is time-dependent
as well. For protons, the accepted ($R_0$, $P_0$) bands lie in the time
interval 30-60 fm$/$c, while for $^3$He, a longer time evolution (at about
60-90 fm$/$c) is needed. Otherwise, much smaller $R_0$ and $P_0$ values should
be chosen. If we further look at the rapidity distributions from both central
and semi-central collisions, it is found that the accepted [$t_{\rm cut}, (R_0,
P_0$)] assemble can provide consistent results for proton yield and collective
flows especially at mid-rapdities, while for $^3$He, the consistency is
destroyed at both middle and projectile-target rapidities.
