Motion of a falling drop with accretion using Newtonian methods

the motion of a falling drop whose mass grows by accretion is studied with newtonian methods to the point of finding the position as a function of time. the equation of motion applied is the equation of motion of continuum mechanics in its eulerian or space formulation. we study three examples of laws of accretion: mass growing linearly with time, mass growing linearly with the surface of the drop and mass growing proportionally to the product of surface and velocity. these examples are sometimes left as exercises, without further discussion, asking only for v(z), or the final velocity. we also show that the solutions have the correct limit of a particle of constant mass in free fall and of such a particle with friction linear in the velocity.