  The physically realistic model of compact stars undergoing gravitational
collapse in $ f(R) $ gravity has been developed. We consider a more general
model $ R + f(R) = R + k R^m $ and describe the interior space-time of
gravitationally collapsing stars with separable-form of metric admitting
homothetic killing vector. We then investigate the junction conditions to match
the interior space-time with exterior space-time. Considering all junction
conditions, we find analytical solutions describing interior space-time metric,
energy density, pressures, and heat flux density of the compact stars
undergoing gravitational collapse. We impose the energy conditions to the model
for describing the realistic collapse of physically possible matter
distribution for particular models of $GR$, $ R+k R^2 $ and $ R+k R^4 $
gravity. The comprehensive graphical analysis of all energy conditions show
that the model is physically acceptable and realistic. We additionally
investigate the physical properties of collapsing stars which are useful to
decipher the inherent nature of such gravitationally collapsing stars.
