Question: I was once asked the following question by a student I was tutoring; and I was stumped by it:

When one throws a stone why does it take the same amount of time for a stone to rise to its peak and then down to the ground?

* One could say that this is an experimental observation; after one could envisage, hypothetically, where this is not the case.
* One could say that the curve that the stone describes is a parabola; and the two halves are symmetric around the perpendicular line through its apex. But surely the description of the motion of a projectile as a parabola was the outcome of observation; and even if it moves along a parabola, it may (putting observation aside) move along it with its descent speed different from its ascent; or varying; and this, in part, leads to the observation or is justified by the Newtons description of time - it flows equably everywhere.
* It's because of the nature of the force. It's independent of the motion of the stone.

I prefer the last explanation - but is it true? And is this the best explanation?

Answer: 
(Non-kinematics math attempt but just some principles)

It is a partial observation in that 

* It hits the ground with same speed.
* Angle by which it hits the ground is the same (maybe a direction change)
* It takes equal time to reach to the peak and then hit the ground

They are equally strange coincidences. Which are more fundamental? Consider the following setup: There are two smooth surfaces of a mountain with different angles. You kick a ball from the ground along a surface towards the hill top so that it just reaches the peak. Now if it rolls back from the same surface, all the above entities will be preserved (including the angle as the ball rolls down on the same surface). But if the ball just goes over the peak (with almost zero speed) and rolls down via the other surface, it will reach the ground at in shorter or longer time and with a different angle. But the speed will be the same (conservation of energy!). 

So, once an entity is conserved, there we can find a symmetry. The opposite is also true. If there are symmetries, then some thing must be preserved. I heard this is proved mathematically long ago. 

In your setup you have all the symmetries and it is no wonder you have conserved entities in the process.
**Assuming that the only forces acting on the stone are the initial force exerted by the hand on the stone AND the force of gravity**, we have the following situation:

Immediately after throwing, the stone has a kinetic energy in the x-direction that never changes (**conservation of energy**) and the stone has a kinetic energy in the y-direction that is constantly being depleted by gravity as it rises to its apex. Once the stone is at its apex (vertical kinetic energy = 0), gravity **works** to restore all of the stone's original vertical kinetic energy as it falls back to its original height.

So, the x-direction **kinetic energy** remained constant throughout the flight and the y-direction **total energy (kinetic + potential)** remained constant throughout the flight, independent of each other. This combination causes the symmetry of the situation.

If air resistance and terminal velocity are introduced into the problem, the character of the problem is completely changed. Even a very simple situation as the one you described can become very complex and confusing to most people. Most people would be stymied by the complexity of the math involved even in this simple situation, let alone the complexity of putting a probe into orbit about Earth.