There was a request for other ways of looking at it, so
here goes:
1) Newton's laws, as they are usually understood, refer to
*particles* in a *classical* situation. These are the standard
limitations on the validity of Newton's laws. Under these
limitations, the mass cannot change. So the third term in the
equation of interest is identically zero, and the question of
what is "v" simply does not arise.
2) Perhaps the question intended to ask about "the laws of motion"
generally, as opposed to Newton's laws in the strict sense. So
let us consider the obvious generalizations i.e. follow-up questions.
2a) In a relativistic situation (as opposed to classical) it is a
big job to generalize Newton's laws. There are two inequivalent
ways to generalize the notion of velocity, and various ways to
generalize the notion of mass, plus at least two inequivalent ways
to generalize the notion of force. The recommended approach is to
start with the relativistically-correct equations of motion and
show that they reduce to Newton's law in the classical limit, rather
than starting from Newton's laws and going in the other direction.
2b) There are cases where it is appropriate to consider the mass in
a *region* rather than the mass of a particle... and similarly the
energy and momentum in the region. Examples include the rainy
railcar we discussed last week, rocketry, and others. The methods
for dealing with such situations go under the name of *fluid dynamics*.
This is quite an interesting subject. It is not an elementary
subject. It is about as tricky as anything I know... in some sense
more full of surprises than quantum mechanics. Everybody *thinks*
they know about fluids, based on their everyday familiarity with
air and water, but a real quantitative understanding does not come
cheap.
A useful exercise is to derive Euler's equation. This describes
the momentum balance in a region subject to inflow and outflow of
material, as well as forces such as pressure and gravitation. The
steps are outlined at http://www.av8n.com/physics/euler-flow.htm
(Beware, there are a lot of bogus derivations out there.)
This is the equation that replaces F=ma when we consider regions
(not particles) and neglect viscosity.