I have been wondering this since learning about position, velocity, and acceleration vs time graphs but can't put numbers/equations to it.
I know that acceleration acts to change velocity, shown by the affect gravity has on an object when thrown upward. Similarly, when driving in a car, you speed changes/impacts your displacement. But how does acceleration, then, impact displacement? How would you show this in some sort of quantitative way? How is your average velocity and instantaneous velocity related to your displacement (same idea with the relationship for acceleration)?
For acceleration, obviously, as you speed up, your displacement will rapidly increase as well, right? So, would you write it something like $ s^{2} \propto a $ or is there some other intuitive relationship that can be represented with numbers?
Is it as simple as something like $v = \frac{\Delta s}{\Delta t}$ so is $ v\Delta t \propto \Delta s $?
Similarity, $a = \frac{\Delta v}{\Delta t}$ so $ a\Delta t \propto \Delta v $?
But how would you relate accleration and displacement together? I know they must be linked, but I'm not sure how exactly.
Could I do something like $\Delta t = \frac{\Delta s}{v}$ and $\Delta t = \frac{\Delta v}{a}$ so $v\frac{\Delta v}{a} \propto \Delta s$? But this doesn't provide much insight at a glance...
Another idea, is it possible to relate $\int_0^T \Delta v\cdot dt = s(T)$ (From the 3Blue1Brown integration video) to acceleration? Or even $a = \frac{d^2s}{d^2t} $ maybe? Simply put, how are these three concepts related and linked to each other as well as how do they impact and affect each other?