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Several things will happen, but the general guiding principle can be understood according to Newton's Laws; particularly, that for every action, there is an equal and opposite reaction.
The first thing that happens when the two objects contact each other is that both will begin to compress. This is due to the fact that the kinetic energy of their movement is nowhere near enough to break the bonds that hold the objects together, but it's enough to put some strain on the objects and "bend" their bonds a little.
As the ball and bat compress, they eventually reach an equilibrium point, where all of the ball's kinetic energy has counteracted some of the kinetic energy of the bat, and the ball has a velocity of 0. As this is taking place, both objects are heating up due to the aforementioned stress being put on their bonds. As the ball and bat reach this equilibrium point, the continued force being exerted by the batter is what will govern the final outcome, because the ball has now used up all of its energy.
Meanwhile, the ball will begin to accelerate in the opposite direction of its original path. The amount of acceleration depends on the force being imparted from the bat and the distance over which the two are in contact. The ball will leave the bat when the arc of the batter's swing and the gradual reduction in kinetic energy as it is imparted from the bat to the ball cause the ball to take up more and more of the kinetic energy in this system.
In the end, the energy of the system will be conserved, but some of it will have been converted to heat and sound, diminishing the proportion available to move the ball.
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