Inquiry on Orbital Mechanics I'm an undergraduate (currently studying precalculus) with an interest in physics. Specifically, I'm interested in the relationship between physics and math which seems to occur often. For example, it is my understanding that Newton conceived of calculus due to his need to describe physical problems where quantities were changing by a changing rate (perhaps the position of a mass under a gravitational attraction, but this is just my guess). I've been wondering if there's a mathematical way to resolve an intuitive question I've been having in my mind about orbital mechanics. Specifically, my question is: if we assume earth to be a flat plane of infinite size and the force of gravity is an acceleration of $9.8$ m/s$^2$, is there some horizontal velocity or acceleration where an object dropped from a height $d_0$ above the surface would not ever touch the surface?
My intuitive assumption is that, if earth is idealized to be an infinite flat plane, then the altitude of a particle with respect to time is $d(t)=d_0-4.9t^2, t\ge0$. Intuitively, it follows that no matter how large of a constant horizontal velocity the particle possesses, it will hit the ground when $d(t)=0$. Indeed, this is not the reality of our spherical earth because some horizontal velocity exists where the ground curves away from the particle before it can hit it; the particle's trajectory matches the curvature of the ground underneath it. However, the picture is not so intuitively clear if the particle were to possess an accelerating horizontal velocity. How do I mathematically or logically investigate the question "if a particle is horizontally accelerating some value $a$ m/s$^2$ above a flat plane, $d(t)$ never becomes $0$?" I suppose the question can better be framed as:
"Is the scenario of a particle traveling with horizontal velocity $v$ m/s above a curved surface such that it never touches the surface (our real earth) equivalent to the scenario of a particle traveling with horizontal acceleration $a$ m/s$^2$ above a flat surface such that it never touches the surface (an idealized flat earth)? (Assuming g = 9.8 m/s$^2$.)"
I feel there must be some concepts or questions similar to this somehow... I'd like help finding those resources, if they exist.
 A: One helpful thing here is that we can explicitly solve for the amount of time it will take for it to hit the ground by solving
$$0 = d_0 - 4.9 t_0^2$$
to get $t_0 = \sqrt{\frac{d_0}{4.9}}.$ We don't consider the negative square root here since we aren't looking backwards in time. So now we have an exact number on how long it will take the particle to hit the ground. The horizontal distance traveled in that time will be the product of $t_0$ and the average horizontal velocity between $t=0$ and $t=t_0$. If the average horizontal velocity is finite, then we travel a finite distance before hitting the ground, and so we don't escape the infinite plane. If the horizontal velocity went off to infinity before $t=t_0$, then the particle not be on a point in the plane anymore, but there isn't any actual physical situation where a particle's velocity would tend to infinity, so it's one of these things where physics constrains what types of mathematical functions we consider.
I'm not sure there's a very satisfying answer to your question other than "particles will always hit the ground," but if you continue studying physics then I think you'll enjoy learning about special relativity based on the way that you're thinking about things.
