Answered

how is the law of conservation proved to be accurate in the energy changes that happen when an object falls from a table to the floor

Answer :

At the top of the table the object has a kinetic energy of zero but a gravitational potential energy of mgh. When the object drops and reaches the bottom, its potential gravitational energy is 0, because it all was converted and conserved in kinetic energy that is [tex](1/2)mv^{2}[/tex] 

Then of course once it hits the ground the energy is lost to the surroundings. 

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