Here today, I've got an example of how it's useful to understand (fundamentals) of Differentiation in early physics.
Mechanical 'laws' of motion say that when you hold a bowling ball in one hand, and a feather in the other hand, evenly, then both will drop to ground the same way, when released at the same time.
Now I worked this out in my home, no books and no WIKI...
A Quiz question:
Please describe the forces on the two objects, as you hold them. The downward force is considered 'static' just simply because there isn't any movement, relative to, say, an Earth frame of coordinates.
I'm better off using a baseball, and with the bowling ball (measuring) say 20 times the weight, of the baseball in your other hand. Assume this question wants a narrative answer, rather than strictly numbers.
The classic equation, likely co-invented by others besides Newton, classic equation is:
F = M A
Now, as I will be teaching soon, that equation reflects result of doing TWO separate differentiations. It's a hierarchy where the first differentiation is performed on the position (equation or function of time elapsed ), while the second differentiation gets you to that acceleration equation, above.
O.K. then, so let's go ahead and drop both test objects, at same time, watching them drop and hit ground, all the whole staying abreast of each other. Must do this in a near-vacume of course,
but say you've got the rocketry...
It's easy to point out that the 'heavy' bowling ball will likely need more FORCE from gravity, vs. the 20X lighter baseball. But since the movement or falling behaviours are same you can conclude: The bowling ball needed 20X the force, for this equivalent falling characteristics.
Somehow the uniform gravity field is pulling on the two different objects precisely proportional, to each of the weights.
I haven't discussed that force yet, but it turns out to be proportional to any two bodies. For the bowling ball that would be:
Weight of Earth X weight of bowling ball
And, of course, same for a second object dropped; that's Weight of Earth X weight of second object...the baseball.
(Don't get mixed up, as each falling object is separate from the other, as the gravity force operates on each (ball) completely separately).
So, assuming that your static force equation has that multiplication of the two interacting weights (one object, with the Earth, in two parallel trials), you can start to prove or at least verify that formula, which turns out to involve each of the two weights, with some constant in there. It's inescapable, as you will observe that in each separate trial, the different forces work to create identical outcomes, in movement measured relative to elapsed time. The lighter baseball, (as second object), gets a smaller force, meanwhile the heavy bowling ball (again, as second object relative to Earth's mass), will get a larger force...just enough so that the outcome, ACCELERATIONs, and subsequent falling positions is the same.
Only conclusion is that ANY weight object will fall just the same as another.
You just have to get air friction out of the test, like on the moon.
Q.E.D.
thanks for reading!