Note mathematics are models of the real world.
What makes quadratic equations nice is they either have a maximum or minimum. So for many problems we are trying to solve; our systems have maxima and minima. a quadratic representation lets us model such a system's behavior close to the maximum or minimum value.
In trying to help my students to understand the motivation for various models, I will sometimes show curve fitting (done via software) to real data. This illustrates real data can be noisy and that our models may not be exact but can be good enough to predict from the data.
As an aside MAA has a nice textbook - Functions, Data, and Models An Applied Approach to College Algebra, that explains from this perspective, motivating the algebra from how we have to apply it to real world data.
So when you look back at the flight of the ball the quadratic form happens naturally, a ball will rise to a maximum point fighting gravity until gravity stops its upward motion and then fall towards the ground with gravity. Since gravity works in the up and down direction (y or h) in the equation this motion will be symmetric around that point. This language is also how we would describe a graph of a quadratic equation. If you progress to calculus you can derive the quadratic nature of this system by knowing that the acceleration due to gravity is constant. But that is another discussion for a later time.
Here are some resources about the thought process of choosing the model.
Once a model is selected you use the tools of algebra to bring out all the predictive power of that model.
CK-12 Linear, Exponential, and Quadratic Models
Monterey Institute Algebra Course
Math is Fun Examples
Note many of the resources you find will start with the flight of a projectile because this is a real life experience that is common to many people so it is used to bridge conceptually from the formula to application.
To Specifically answer your questions about the historical motivation:
University of Virginia Math History
Mathnasium a nice summary timeline