Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed from one form to another.
Definition
The law of conservation of energy says that energy can never be made from nothing or destroyed; it can only change from one form into another or move from one object to another. Stated for a system, the total energy of an isolated system remains constant: energy can be transferred between objects and transformed among forms (kinetic, potential, thermal, chemical, electrical, light, sound), but the total before any process equals the total after, and energy that seems "lost" has usually become thermal energy. Formally, $dE_{\text{total}}/dt = 0$ for an isolated system, where $E_{\text{total}}$ includes mechanical, thermal, chemical, electromagnetic, and nuclear forms, and for a system exchanging energy with its surroundings, $\Delta E_{\text{system}} = Q + W$, the first law of thermodynamics. By Noether's theorem, energy conservation follows from the invariance of physical laws under time translation. In relativity, mass is included through $E = mc^2$, so mass-energy is the conserved quantity.
Example
When you rub your hands together, the energy of motion becomes heat; the motion energy did not disappear, it changed form and warmed your hands. A $1\ \text{kg}$ ball dropped from $2\ \text{m}$ starts with $19.6\ \text{J}$ of potential energy but bounces back to only $1.5\ \text{m}$, or $14.7\ \text{J}$; the missing $4.9\ \text{J}$ became heat and sound during the bounce, and nothing was destroyed. On a larger scale, a $1{,}200\ \text{kg}$ car moving at $25\ \text{m/s}$ that brakes to a stop turns its $\tfrac{1}{2}(1200)(625) = 375{,}000\ \text{J}$ of kinetic energy into thermal energy in the brake discs. If the discs have total mass $8\ \text{kg}$ and specific heat $460\ \text{J/(kg}\cdot^\circ\text{C)}$, their temperature rises by about $375000 / (8 \times 460) \approx 102^\circ\text{C}$ before cooling begins.
Key Insight
If some energy seems to vanish, look for heat, sound, or light; it went somewhere, because energy never just disappears. This law is why perpetual-motion machines are impossible: a machine cannot put out more energy than it takes in, and friction guarantees it puts out a little less. Energy is conserved but not preserved in usefulness. Thermal energy spread across the environment can no longer be gathered to do work, which is the second law's domain: energy quantity is fixed, but energy quality degrades.