Question

Earth's gravity accelerates objects at 9.8 m/s2. When the flying object reaches the top of its arc (even if it is not pictured), it has a vertical velocity of 0 m/s. Ignoring air resistance, what would be the vertical velocity of the object after 1 second?

Answer 1

You haven't told us anything about the toss or the launch,
so we have to assume that you're talking about 1 second
after the top of its arc.

At the top of its arc, the object stops rising and starts falling. 
One second after that moment, it has fallen 4.9 meters from
the top of its arc, and is falling at a speed of 9.8 m/s.
It's vertical velocity is 9.8 meters per second straight down.

Answer 2

The motion of falling objects is the simplest and most common example of motion with changing velocity. The early pioneers of physics had a correct intuition that the way things drop was a message directly from Nature herself about how the universe worked. Other examples seem less likely to have deep significance. A walking person who speeds up is making a conscious choice. If one stretch of a river flows more rapidly than another, it may be only because the channel is narrower there, which is just an accident of the local geography. But there is something impressively consistent, universal, and inexorable about the way things fall.