Why can a ball tossed up object free fall upwards? I've found many websites that say that a ball tossed upward would be free falling upwards, but none of them say why. If the definition of free fall is an object that is falling while only under the force of gravity alone, then the force you are exerting on the ball when you toss it up rules it out from free fall, right?
 A: In common parlance, saying that an object is "falling" implies that the object's velocity is in the direction of the ground.  However, the phrase free fall is defined as applying to an object which has no forces acting on it except for gravity, a definition that has nothing to do with velocity.
If you throw a ball 3 meters above your head, your hand stops applying a force to the ball at the moment the ball leaves your hand.  At that moment, there are no forces acting on the ball any more except for gravity, so it meets the definition of being in "free fall", even though at that point the direction of the ball's velocity is away from the Earth, and hence the ball isn't "falling" as that word is used in common parlance.
To answer your follow-up question, no, the ball doesn't "carry force with it" after it leaves your hand.  After the ball leaves your hand, the ball has kinetic energy and momentum and an upward velocity due to the force that had been on the ball, but there is no force acting on the ball any more.
A: 
then the force you are exerting on the ball when you toss it up rules it out from free fall, right?

Your hand has to be in contact with the ball to exert a force on it. As soon as you let go, you can't provide a force to the ball anymore.
Since you can no longer provide a force to the ball, the only force left on it is gravity, so it's in free-fall. The net acceleration on the ball is now towards the ground. (There are some other forces due to friction present, like drag, but we'll ignore those and pretend the ball is falling in a vacuum.)
The ball's velocity is still upwards for a while, but under the relentless acceleration of gravity, it will slow until it reaches zero at the top of its arc, and then head towards the ground.
The word "fall" in "free-fall" is something of a misnomer. An object need not be moving towards the ground to be in free fall, only accelerating towards the ground under the influence of gravity. For example, these are all in free-fall:


*

*A bullet fired straight upwards

*A satellite in stationary orbit

*A cannonball shot from a cannon

A: Yes because when you put force when the object is falling it is no longer free falling. Free falling is when you have a object with nothing acting upon it making it free fall.
