  A model which describes the time-dependent CP formalism in $D^0$ decays has
recently been proposed. There it has been highlighted a possible measurement of
the angle $\beta_c$, in the charm unitarity triangle, using the decays $D^0\to
K^+ K^-$ and $D^0\to \pi^+ \pi^-$, and a measurement of the mixing phase
$\phi_{MIX}$. The same method can be used to measure the value of the parameter
$x$, one of the two parameters defining charm mixing. We numerically evaluate
the impact of a time-dependent analysis in terms of the possible outcomes from
present and future experiments. We consider the scenarios of correlated $D^0$
mesons production at the center of mass energy of the $\Psi(3770)$ at Super$B$,
uncorrelated production at the center of mass energy of the $\Upsilon(4S)$ at
Super$B$ and Belle II, and LHCb. Recently a hint of direct CP violation in
charm decays was reported by the LHCb collaboration, we estimate the rate of
time-dependent asymmetry that could be achieved using their available data, and
we generalise the result for the full LHCb program. We conclude that LHCb is
already able to perform a first measurement of $\beta_{c,eff}$, and slightly
improve the present constraints on the parameters $x$ and $\phi_{MIX}$. A more
precise determination of $\beta_{c,eff}$, $\phi_{MIX}$ and $x$ will require a
larger data sample, and most probably the cleaner environment of the new high
luminosities $B$-factories (both Super$B$ and Belle II) will be needed. We show
that Super$B$ will be able to measure $\beta_{c,eff}$ and $\phi_{MIX}$ with a
precision of $1.4^o$ and improve the precision on $x$ by a factor of two.
