Compare the potential energies of a body on the surface of the Earth and on the surface of the Moon, Accept that the acceleration of free fall on the Moon is six times less than on the Earth, and the radius of the Moon is three times less than the radius of the Earth.
Decision:
To compare two potential energies, they must be counted from the same level. In this case, it is most convenient to take an infinitely distant point as the zero level. Then
$U_{e}=-G \frac{mM_{e}}{R_{e}}=-mg_{e}R_{e}$,
$U_{m}=-G \frac{mM_{m}}{R_{m}}=-mg_{m}R_{m}$.
We see that the potential energy of a body on the Moon is greater than on the Earth by the amount of
$U_{m}-U_{e}=m(g_{m}R_{m}+g_{e}R_{e})=\frac{17}{18}mg_{e}R_{e}$.
Decision:
To compare two potential energies, they must be counted from the same level. In this case, it is most convenient to take an infinitely distant point as the zero level. Then
$U_{e}=-G \frac{mM_{e}}{R_{e}}=-mg_{e}R_{e}$,
$U_{m}=-G \frac{mM_{m}}{R_{m}}=-mg_{m}R_{m}$.
We see that the potential energy of a body on the Moon is greater than on the Earth by the amount of
$U_{m}-U_{e}=m(g_{m}R_{m}+g_{e}R_{e})=\frac{17}{18}mg_{e}R_{e}$.