Conservation of Mechanical Energy

Mechanical energy is the sum KE + PE, where KE = one half m v² and PE depends on what's storing it: m g h for gravity, one half k x² for a spring. If the only forces doing work are conservative ones, like gravity or an ideal spring, that sum is locked. Nothing new appears and nothing leaks out. As an object falls, height drops so gravitational PE drops, and that lost PE shows up exactly as gained KE. Run the motion backward and the trade reverses. The roller coaster never climbs back to its starting height because real tracks have friction and air drag, nonconservative forces that quietly siphon mechanical energy into heat and sound. In a frictionless idealization, it would return to exactly the same height, no higher, no lower.

The real payoff is that you skip the forces entirely. Instead of integrating Newton's second law through a curved slide or a changing spring force, you just compare energy at two points: initial KE plus initial PE equals final KE plus final PE. This is why conservation of energy sits right after work and potential energy in this unit: work is how energy moves, potential energy is the stored version tied to position, and conservation is the bookkeeping rule that makes the whole system solvable without ever writing F = ma. It's also the bridge into problems with nonconservative forces, where you'll track energy lost to friction as negative work done, rather than assume the total stays put.