  The paper is devoted to the motion of a body in a fluid under the influence
of gravity and drag. Depending on the regime considered, the drag force can
exhibit a linear, quadratic or even more general dependence on the velocity of
the body relative to the fluid. The case of quadratic drag is substantially
more complex than the linear case, as it nonlinearly couples both components of
the momentum equation. Careful screening of the literature on this classical
topic showed that, unexpectedly, the solutions reported do not directly provide
the particle velocity as a function of time but use auxiliary quantities or
apply to special cases only. No explicit solution using elementary operations
on analytical expressions is known for a general trajectory. After a detailed
account of the literature, the paper provides such a solution in form of a
ratio of two series expansions. This result is discussed in detail and related
to previous approaches. In particular, it is shown to yield, as limiting cases,
certain approximate solutions proposed in the literature. The solution
technique employs a strategy to reduce systems of ordinary differential
equations with a triangular dependence of the right-hand side on the vector of
unknowns to a single equation in an auxiliary variable. For the particular case
of quadratic drag, the auxiliary variable allows an interpretation in terms of
canonical coordinates of motion within the framework of Hamiltonian mechanics.
Another result of the paper is the extension of the solution technique to more
general drag laws, such as general power laws or power laws with an additional
linear contribution. Furthermore, generalization to variable velocity of the
surrounding fluid is addressed by considering a linear velocity profile for
which a solution is provided as well. Throughout, the results obtained are
illustrated by numerical examples.
