Gravity

The potential energy of an object experiencing a gravitational force (only) is:

$$U_g = - \int_0^y (-mg)dy = mgy$$ (3.36)

More generally, we write this as:

$$U(y) = mgy + U_0$$ (3.37)

where $U_0$ is an arbitrary constant that sets the zero of gravitational potential energy. You will find it very helpful to choose a coordinate system (or set the zero of potential energy) in such as way as to make the problem as computationally simple as possible. That usually means arranging it so that either the initial of final potential energy is zero.