Gravitational Potential Energy: Changing Kinetic Energy

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Physics notes:
Note: The starting point for this problem could be either of two choices.

FIRST METHOD OF CHOICE:
Note 1.
Start with the expression giving the gravitational PE of two bodies of mass M and m, separated by center to center distance r, which is expressed by:
(1) P.E. = - G M m/r
Note that this quantity is negative because the P.E. is defined to be zero when the separation r is infinite. Therefore if r increases from some initial value, then for the P.E. to approach zero, it has to initially be a negative value.
Note 2.
By conservation of energy, as the projectile ...