Apparently the answer "the 2 moments coincide" was not a valid answer.
Lets consider the event ($x_0, t_0$) when the car was last stationary and ($x_1, t_1$) when the car was first moving. If $t_1 = t_0$ and $x1>x_0$ then we have $v = \delta x/\delta t = \infty$ which violates special relativity. If $x1=x_0$ then we have $v = \delta x/\delta t = 0/0$ which is indeterminate, but it also implies that event ($x_0, t_0$) is the same as event ($x_1, t_1$) and that they are in fact the same event and no motion has occurred. This is perhaps what your teacher was getting at when he said ""the 2 moments coincide" is not a valid answer. This paradoxical situation arises if we consider time and space to be infinitely divisible.
Calculus tells us we can assign an exact velocity value to an exact instant of time. That means we can ascribe a velocity of zero at instant zero and a non-zero value of velocity at a later instant. To say there is a zero interval between time zero and the next instant is to imply the velocity at time zero is both zero and non zero, which is illogical. If we assume a time $t_1$ which is the smallest time greater than $t_0$, we can always halve that value and find a smaller time. Let's assume at some small finite time, the time interval $\delta t_n$ is described by some fraction $\frac1{2^{n}}$, where n is an integer. As n goes to infinity, the interval goes to zero, but implicit in this assumption is that we can arrive at infinity by continuously doubling a number, which is of course nonsense.
In physical reality, one explanation is the Heisenberg uncertainty principle. This principle tells us the more certain we are about the momentum of a particle the less certain we can be about its position and vice versa. This means if we are absolutely certain a particle is stationary, we have a huge uncertainty in the particles position at time zero and if we are absolutely certain of a particle's location at time zero we have a huge uncertainty in its velocity. This make it impossible to say a particle is stationary at some exact instant, so the question in the title becomes almost impossible to define.
Almost all the paradoxes ascribed to Zeno are based on what would happen if we assume space and time are infinitely divisible. The Achilles and the tortoise paradox and the dichotomy paradox (and the question the OP) can be boiled down to whether or not the infinite sum $$\frac12 +\frac14+\frac18+\frac1{16}+ ...$$ is equal to 1. Almost universally, the modern interpretation is the infinite sum is exactly equal to one. It can be shown that for a finite series of terms the sum can be written as:
$$ S_n = 1 +\frac{1}{2^{n}}$$
For $n = \infty$ we get:
$$ S_{\infty} = 1 +\frac{1}{2^{\infty}} $$
and as n approaches infinity, the term
$ \frac{1}{2^{\infty}} $ approaches 0 and so $S_n$ approaches 1.
Is this the same thing as saying the gap between one and the previous term is exactly zero and $S_n$ is exactly one?
The concept of surreal numbers would beg to differ. The Wikipedia article states (paraphrased) that infinitesimal numbers are smaller in absolute value than any positive real number. In other words there is a surreal number ($1/\infty$) between the smallest real number and zero and therefore the gap between the smallest real number and zero is not zero and this is basically the answer to 'What happens between the last moment the car is at rest and the first first moment the car moves?'. Also note that in the set of surreal numbers, the number $1/\infty$ is not equal to zero. In fact in this Numberphile video, Donald Knuth states there an infinite number of surreal numbers between any two real numbers, but what does he know?
Should we dismiss surreal numbers out of hand? After all, how can we discuss what is between the smallest real number and zero if we cannot define what the smallest real number is? Whatever we define it to be, we can always divide it half again and find a smaller real number. If I wanted to be awkward I could say, first tell me what the smallest positive real number is, before we discuss what is between the smallest
positive real number and zero.
We can note that all of Zeno's paradoxes that involve motion or time, e.g. the Achilles and the tortoise paradox, the dichotomy paradox or even the arrow paradox, can be resolved if we assume there is an minimum indivisible discrete unit of time, possibly much smaller than the Planck interval. Unfortunately we are unable to probe down to that level and so there is no measurement we can perform to support or refute that idea.
Unfortunately I cannot provide a definitive answer to the OP and my personal belief is that there is no definitive answer, but I should stress that the majority accepted answer is that there is a definitive answer to the OP question (and Zeno's paradoxes) which is based on the premise that the infinite sum 1/2+1/4+1/8+1/16+... is exactly equal to 1 and that $1/\infty$ is exactly equal to zero, but to me, this implies that the gap between the smallest real number and zero is exactly zero. To me, the issue is not clear cut.
