Gravitational potential energy

As we saw above, the work done to go from one point to another in a constant gravitational field is independent of path, and in fact it only depends on the difference in vertical height. So to go between two points ${\bf r}_i$ and ${\bf r}_f$ we can choose a straight line path. The work is just ${\bf F}\cdot \Delta {\bf r}$, but ${\bf F}~=~ -mg \hat j$. But $\Delta {\bf r}\cdot \hat j ~=~ \Delta y$, the difference in the vertical coordinates. So we have the work is just $-mg(y_f -y_i)$. We can define the potential energy uniquely by saying we want $U ~=~ 0$ at $y ~=~ 0$. So from the definition of potential energy, eq. 1.16 we have have

$$U(y) ~=~ -W ~=~ mgy$$ (1.17)