# Proof 36: Gravitational potential energy

Small falling object due to weight

A flower pot falls down from a window sill due to gravity. This i because gravity does work $W_g$ on it, pulling it down with its gravitational force, called weight $w$. The distance over which this work is done is the height $h$, if we pick the reference (the level from which we measure the height) to be the ground.

\begin{align} W&=Fd\qquad\leftarrow \text{Work formula}\\ W_g& =wh \qquad\leftarrow \text{The distance moved is the height } h\\ ~& =mgh\qquad\leftarrow \text{Weight formula, } w=mg \end{align}

This derivation can be redone for any reference. The work done by gravity is what we tend to call gravitational potential energy, since gravity is a conservative force. So, symbolising it $U_g$, we have the formula:

$$U_g=mgh\qquad\blacksquare$$