In a gravitational field, all objects accelerate uniformly with acceleration equal to g = -9.81 m/s (» -10 m/s) We assign a negative number because we normally consider "upward" motion to be positive, and gravity causes a "downward" acceleration.
If you drop something, after t seconds of free fall, its velocity is given by v = gt. For example, if you drop a baseball off a cliff, after 1 second it is moving at -10 m/s; after 2 seconds at -20 m/s; and so on.
The position of the ball is given by x = ½gt2 so that after 1 second it has dropped 5 meters; after 2 seconds it has dropped 20 meters; and so forth.
(In the example, v0 = 0 and x0 = 0.)
If you throw a ball with velocity v0, its velocity at time t is given by v = v0 + gt. For example, if you throw a ball downward with an initial velocity vi of -5 m/s, its velocity after 1 second is -15 m/s; after 2 seconds is -25 m/s; and so forth. The derivation is the same as for v = gt.
The position of the ball is given by x = x0 + v0t + ½gt2 so that in our example (letting x0 = 0) the ball has fallen 10 m after one second, 30 m after 2 seconds, and so forth. The derivation is the same as for x = ½gt2.
A ball thrown upward is a more interesting case. The velocity is given by the same equation, but we need to remember that a velocity in the opposite direction is given a negative number (conventionally we call "up" positive and "down" negative). So if you throw your ball up with a velocity of +5 m/s, after 1 second its velocity is given by
5 m/s + (-10 m/s/s)´(1 s) or -5 m/s--that's 5 m/s down.
The ball's position is given by
(5 m/s)×(1 s) + ½(-10 m/s/s)×(1 s)2 or 0 meters--that's the initial position of the ball!
In general, your ball thrown upward will spend the same time getting back down to where you are that it took going up, and will be moving at the same velocity downward as the velocity with which you threw it up. See for example Figure 1.12.