Potential Energy
Potential energy is stored energy an object has because of its position, shape, or condition, ready to be released as motion or other forms.
Definition
Potential energy is stored energy: an object has it because of where it is or how it is arranged, and that energy is waiting to be let loose. Gravitational potential energy depends on height ($PE = mgh$), elastic potential energy is stored in stretched or compressed springs and bands, and chemical potential energy is stored in the bonds of food, fuel, and batteries; all of them can transform into kinetic energy and back. Strictly, potential energy is defined only for conservative forces, those whose work is path-independent, as the negative of the work done by the force, $\Delta U = -\int \vec{F}\cdot d\vec{r}$, so the force is the negative gradient of the potential, $F = -dU/dx$. Common forms are gravitational $U = mgh$ near Earth's surface and $U = -GMm/r$ in general, and elastic $U = \tfrac{1}{2}kx^2$. Only differences in $U$ are physical; the zero point is a free choice.
Example
A stretched rubber band, a book on a high shelf, and a wound-up toy all have potential energy, and when you let go, the stored energy turns into motion. A $0.2\ \text{kg}$ arrow held at the top of a $30\ \text{m}$ tower has $PE = 0.2 \times 9.8 \times 30 = 58.8\ \text{J}$, which becomes kinetic energy as it falls. A spring with $k = 400\ \text{N/m}$ compressed $0.10\ \text{m}$ stores $U = \tfrac{1}{2}(400)(0.10)^2 = 2.0\ \text{J}$. Released against a $0.05\ \text{kg}$ ball, it gives $\tfrac{1}{2}(0.05)v^2 = 2.0$, so $v = \sqrt{80} \approx 8.9\ \text{m/s}$.
Key Insight
Potential energy is invisible until it is released; a boulder at the top of a hill looks calm, but it is storing a lot of energy. A roller coaster's first hill is always the tallest because the car never gets more energy than it stored at the top, so every later hill must be lower. Objects tend to move toward lower potential energy, since force points downhill on the potential-energy curve. Stable equilibrium sits at the bottom of a potential well, and unstable equilibrium at the top of a hump.