Archimedes principle of buoyancy has to do with the displaced water being equal to the force of the object's departure from the water.
He found that anything with a rounded bottom and air above it will float.
The buoyant force is the upward force on an object immersed in or floating on a fluid. If the object weighs less than the buoyant force, it will float on top of the fluid. If it weighs more, it will sink.
The quote above comes from the second link. These are both short quick reads and worth a look.
Archimedes' principle states that an object placed in a liquid experiences a buoyant force equal to the weight of liquid it displaces. This force pushes the object upward towards the surface of the liquid.
An inflated beach ball completely immersed in pool of water will displace water equal to the volume of the beach ball. Therefor according to Archimedes' principle it is clear that the beach ball will experience a buoyant force equal to the weight of water equal to the weight of water equal to volume of ball. This force is much more than the weight of the ball. Thus the ball can be kept immerses in water only by some devise holding it down. Bus as soon as the ball is released it will be pushed out of water under the influence of the buoyant force less weight of the ball. As the ball moves up to the surface of the pool its velocity will increase, and as it emerges out of water, it will continue to move upward in air because of kinetic energy of the velocity it has acquired. Once our of water there is no buoyancy force acting on the ball that pushes it upwards. Rather there is gravitational force that pulls the ball down. Under the influence of this gravitational pull the ball will slow down as it rises above the water surface till its velocity becomes zero. At this point the ball is at its highest point. After this the ball will begin to fall down.
Archimedes, the pholsopher scientist found that an object that floats or submerged in a liquid looses its weight on account of the upward buoyant force on the object by the liquid. The upward force is equal to the weight of the liuid that object dispells. Therefore, the lighter object (less dense than the liquid or water) , if sumerged, gets more upward thrust than the weight of the object. So the object is pushed up and the object floats , only partly bing submerged and partly above the level of the liquid. If the object has a density higher than that of the liquid, then it sinks in the liquid but its apparent weight is less by the wieght of the equal volume of the liquid.