Why does newtons second law involve second derivative of position?

Question

As in, why is newton's second law(for constant mass systems),
$$ F= m frac{d^2 x }{dt^2}$$
and not something of the sort like
$$ F= m frac{d^3 x }{dt^3}$$
Why is it that sum of all force can be equated to mass times second derivative of position? Like in all cases the right side of the $F=ma$ equation is same.
My attempts at solving this question:
I saw this stack  which led me to this other stack and also this other one, both of which requires a lot more mathematical context than I know already (Lagrangian and Hamiltonian formalism). So, I look for a simpler explanation using mainly physical principles supplemented with nothing more than basic vector calculus to explain why we can equate the sum of all forces as mass times the second derivative of position.

Vercassivelaunos · Accepted Answer

Newton's actual second law is not $F=ma$, but
$F=dot p.$
And that's more or less what defines force*. Based on the observation that momentum $p=mv$ of an object only changes when it interacts with something, we can see that the change in momentum is an interesting quantity. And since change in momentum is intuitively associated with what laymen would call force, we just call it force as well. So that's a definition, $F:=dot p$. And since usually the mass of an object doesn't change, this becomes
$F=dot p=underbrace{dot m}_{=0}v+mdot v=mdot v=ma.$
It's not true for rockets, though, since they shed mass to accelerate, so $dot mneq0$. But that's just an interesting side fact.
To summarize: It can be observed that $mv$ only changes due to external interaction, which makes its change an interesting quantity, and we call that quantity force. That's why force is naturally just the derivative of $mv$, which usually reduced to $ma$.
*Not going into the intricacies of static forces

user137289 · Answer

Why look further than the first law for a simple explanation?

An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

A force causes a change in velocity. Velocity is the first derivative of position. If the second derivative of position is not zero, the velocity changes, so there must be an unbalanced force.

Adam Herbst · Answer

To add a little background on top of the current answers:
As I understand it, prior to Galileo and Newton people believed that you couldn't even have velocity without force; that is, objects maintain their position unless forced.  That's because they were used to objects sitting on the ground, which will not budge unless you push them, and normally stop moving the instant you let go.
At some point, someone (I think Galileo?) realized an object can move without being pushed, for example on ice.  After you let go, it keeps moving for a while.  Yet the ice does not seem to be pushing it, so it must be slowing it down instead.  Which means that an unforced object actually tends to maintain its velocity, not its position.
That inspired Newton to formulate a law saying that force is what causes changes in velocity.
But if we replaced acceleration by its derivative, that would be saying that force is what causes changes in acceleration.  In other words, an object will maintain its acceleration unless forced to change it.  But that would mean that, for example, if you push an object until it's got a certain acceleration, and then let go, it will keep getting faster at that same rate unless something forces it to start slowing down, even if it's just drifting through empty space.  That would be a strange world, and it's not what we observe.

Why does newtons second law involve second derivative of position?

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