  The family of exactly solvable potentials for Newton's equation of motion in
the one-dimensional system with quadratic drag force has been determined
completely. The determination is based on the implicit inverse-function
solution valid for any potential shape, and hence exhaustive. This solvable
family includes the exponential potential appearing in the Toda lattice as a
special limit. The global solution is constructed by matching the solutions
applicable for positive and negative velocity, yielding the piecewise analytic
function with a cusp in the third-order derivative, i.e., the jerk. These
procedures and features can be regarded as a generalization of Gorder's
construction [Phys. Scr. 2015, {\bf 90}, 085208] to the energy-dissipating
damped oscillators. We also derive the asymptotic formulae by solving the
matching equation, and prove that the damping of the oscillation amplitude is
proportional to $ t^{-1} $.
