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: 
I think its because both halves of a projectile's trajectory are symmetric in every aspect. The projectile going from its apex position to the ground is just the time reversed version of the projectile going from its initial position to the apex position.
A lot of things have to hold to get that symmetry. You have to neglect air resistance. You either have to throw it straight up, or the ground over there has to be at the the same altitude as the ground over here. You have to through it slowly enough that it comes back down (watch out for escape velocity)

But if you have that, then the simplest explanation in the simplest case is that the acceleration is constant. So you keep going up until you have zero (vertical) velocity, which takes time $v\_0/g$, during which you travel up a certain distance. On the way down, the acceleration is still constant, so now the velocity becomes more down over time, but that just makes it the time reverse (when looking at the vertical component alone) so it takes an equal amount of time.

If you want to throw it high enough that the acceleration starts to vary, an energy argument is quicker, though you can study the kinematics on the way back down as the time reverse (gaining just as much as you lost before, in just as much time, going just as far, etc.).