Potential energy for near Earth gravity

For small height changes, gravitational potential energy can be computed using

${\displaystyle U_g=mgh,}$

where m is the mass in kg, g is the local gravitational field (9.8 metres per second squared on earth), h is the height above a reference level in metres, and U is the energy in joules.

In classical physics, gravity exerts a constant downward force F=(0, 0, F_z) on the center of mass of a body moving near the surface of the Earth. The work of gravity on a body moving along a trajectory r(t) = (x(t), y(t), z(t)), such as the track of a roller coaster is calculated using its velocity, v=(v_x, v_y, v_z), to obtain

${\displaystyle W={\int <!--\; \int \; -->}_{t_1}^{t_2}\mathit{F}\cdot <!--\; \cdot \; -->\mathit{v}\mathrm{d}t={\int <!--\; \int \; -->}_{t_1}^{t_2}{F}_z{v}_z\mathrm{d}t=F_z\mathrm{\Delta <!--\; \Delta \; -->}z.}$

where the integral of the vertical component of velocity is the vertical distance. The work of gravity depends only on the vertical movement of the curve r(t).