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[champlooTTV]

Okay, when you solve for the Force of something, you're solving for it at a specific moment... you're not solving an average. So, when you solve F=(ma)d/dt you will pull m out as a constant, giving you F=m((a)d/dt). You're talking about including the change of mass in a scenario... If that's what you were looking for, then the Force (F) would be notated with an integral before the F AND m would have to be changed to (dm) to represent (dm/dt) (change of mass over time).

tldr; Amamin is right.
source: high school physics and 10 years as a mechanical engineer.

[Bunny]

Okay, when you solve for the Force of something, you're solving for it at a specific moment... you're not solving an average.

When you solve for the "Force of something", you are attempting to find the force required in order to accelerate a body to a given state. This has nothing to do with averages.

So, when you solve F=(ma)d/dt you will pull m out as a constant, giving you F=m((a)d/dt).

Incorrect. The generic form for differentiation of a multiplication is given by:

This is, as described, for a case where one cannot simplify the expression by treating mass as constant when its changing.

You're talking about including the change of mass in a scenario... If that's what you were looking for, then the Force (F) would be notated with an integral before the F AND m would have to be changed to (dm) to represent (dm/dt) (change of mass over time).

Newton's second law of motion does not describe a point in time where you would calculate the forces therein, because it's an inertial equation and it holds true as a function of time, hence derivative over time.
Adding an integral before external forces in the equation has nothing to do with the law itself, rather, it is a way of calculating work done over time by a force, exerting acceleration on an object.
If you were to somehow decide you wanted to integrate over the forces, youll be able to find the final velocity of the object given initial conditions.

In his original scripts, Newton developed his law with the idea that forces applied to a body are equal to the change of momentum over time. He did not simplify the expression to the known F=m*a that most highschool students know today, because it would simply not be inclusive of every case. This, for example, happens when you study flow dynamics and realize that internal body forces, external forces, conservation of mass, momentum and energy need to be given in their most generic state, and based on assumptions, simplified to known terms.

I frankly don't understand why I need to convince someone with "10 years mechanical engineer" experience who should've taken the basics physics 101 course and probably many more physics courses due to the nature of the degree and have seen this for themselves, the phrasing of Newton's second law of motion.