Clockclocle said:
The instantaneous velocity at time a is defined as derivative of motion function f(t). It is not similar to average velocity in an interval of time. From the Newton law. If an object is at rest, we must exert a force to make it move, assume that there is no friction. Depend on the weight of object even though we exert enough force it won't move if we do not stay exert on it enough time then the object must have a small interval of time to change their status of motion. In reality we can't make things change "instantaneous", so I guess in that small interval of time when the object change their status of motion the velocity of the object is the same on all the interval and equal to f'(a)
This is more of a mathematics question than a physics question. Or, perhaps, more about how we join physics with mathematics in a "model". There is no way to do justice to the subject matter in a brief post.
The model you seem to propose is that trajectories are composed of lots of little straight line segments. That once enough acceleration has built up, the particle suddenly snaps from one straight line to the next. The idea is that
if we make the straight line segments short enough, this model is indistinguishable from reality.
I agree. This model works. It is pretty much the basis for
Riemann integration or for the epsilon delta definition of a derivative. It is also how we run numerical simulations with the
Euler method.
[You should know that the Euler method is about as primitive as it gets. You quickly want to improve it -- perhaps by pretending that the particle gets half of its delta v at the beginning of the interval and the other half at the end, thus eliminating some bias]
In mathematics, we phrase things differently. Instead of talking about "short enough", we get down to the nitty gritty about tolerances (the epsilon) and how short you need an arc to be to attain that tolerance (the delta). As a result, we can talk about the "limit" that is approached to within every tolerance you can choose if you can look as closely as you please.
Often, we can solve
differential equations to obtain an exact formula for the motion of a particle under a particular force law. Sometimes we cannot solve the equations. But we can still prove that an exact and continuously differentiable trajectory exists -- one which does not take the form of a bunch of finite straight-line segments.
In summary: Yes, the short straight segment model works. Approximately. But a smoothly curving mathematically perfect trajectory also exists, even if we cannot always write down a formula for it or measure exactly enough to distinguish between the approximation and the mathematical ideal. No, we do not believe that particles actually follow short straight-line trajectories, even though experiment is silent on the question.