  We study the decays of $\Lambda_b \to \Lambda_c(\to B_n f) \ell ^-
\overline{\nu}$ with $\ell = e, \mu, \tau$, where $B_n$ and $f$ are the
daughter baryons and the rest of the particles in $\Lambda_c$ cascade decays,
respectively. In particular, we examine the full angular distributions with
polarized $\Lambda_b$ and lepton mass effects, in which the time-reversal
asymmetries are identified. We concentrate on the decay modes of $\Lambda_b \to
\Lambda_c(\to p K^- \pi^+) \ell ^- \overline{\nu}$ to demonstrate their
experimental feasibility. We show that the observables associated with the
time-reversal asymmetries are useful to search for new physics as they vanish
in the standard model. Explicitly, we find that they are sensitive to the
right-handed current from new physics, and possible to be observed at LHCb.
