  In this note we explore a non-static spacetime in quantum regime in the
background of $f(R)$ gravity. The time dependent Vaidya metric which represents
the spacetime of a radiating body like star is studied in an energy dependent
gravity's rainbow, which is a UV completion of General Relativity. In our quest
we have used gravitational collapse as the main tool. The focus is to probe the
nature of singularity (black hole or naked singularity) formed out of the
collapsing procedure. This is achieved via a geodesic study. For our
investigation we have considered two different models of $f(R)$ gravity, namely
the inflationary Starobinsky's model and the power law model. Our study reveals
the fact that naked singularity is as good a possibility as black hole as far
as the central singularity is concerned. Via a proper fine tuning of the
initial data, we may realize both black hole or naked singularity as the end
state of the collapse. Thus this study is extremely important and relevant in
the light of the Cosmic Censorship hypothesis. The most important result
derived from the study is that gravity's rainbow increases the tendency of
formation of naked singularities. We have also deduced the conditions under
which the singularity will be a strong or weak curvature singularity. Finally
in our quest to know more about the model we have performed a thermodynamical
study. Throughout the study we have obtained results which involve deviation
from the classical set-up. Such deviations are expected in a quantum evolution
and can be attributed to the quantum fluctuations that our model suffers from.
It is expected that this study will enhance our knowledge about quantization of
gravity and subsequently about the illusive theory of quantum gravity.
