You have entered a rabbit hole that kept physicists and philosophers arguing for ages.
The desciption you gave of a car at rest is a classical description in terms of Newton's mechanics. If we assume continuous values for position and velocity, the question of "between moments" becomes ill-defined, since there is no positive real number "next to 0", as others pointed out.
Quantum mechanics, however, gives a different description, where particles do not have position and momentum vectors of infinite precision. Even the lowest energy state is not exactly zero, some positive amount of zeropoint energy remains. We also can't cool particles down to 0 Kelvin and stop them for the same reason. In essence, nature solves this conundrum by making sure nothing is ever at rest. Everything is vibrating all the time.
Respect to your professor for giving young minds a tough nut to crack. Respect to you for still pondering it.