Question
Thu March 22, 2012 By: Pri

show that the sum of kinetic & potential energy is always conserved in the case of the freely falling body under gravity(with air resistance neglected)from a height h by finding it(a)the body is at top,(b)the body has fallen a distance x,(c)the body has reached the ground.

Expert Reply
Thu March 22, 2012
Consider a mass  which is falling vertically under the influence of gravity. We already know how to analyze the motion of such a mass. Let us employ this knowledge to search for an expression for the conserved energy during this process. (N.B., This is clearly an example of a closed system, involving only the mass and the gravitational field.).
 
. Suppose that the mass falls from height  to , its initial velocity is , and its final velocity is . It follows that the net vertical displacement of the mass is . Moreover,  and . Hence, the previous expression can be rearranged to give  

 

The above equation clearly represents a conservation law, of some description, since the left-hand side only contains quantities evaluated at the initial height, whereas the right-hand side only contains quantities evaluated at the final height. Now, let us define the kinetic energy of the mass,  

 

and the gravitational potential energy of the mass,  

 

Note that kinetic energy represents energy the mass possesses by virtue of its motion. Likewise, potential energy represents energy the mass possesses by virtue of its position.

 

Here,  is the total energy of the mass: i.e., the sum of its kinetic and potential energies. It is clear that  is a conserved quantity: i.e., although the kinetic and potential energies of the mass vary as it falls, its total energy remains the same.
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