Conservation of Energy

Calculus 3 · Axiom Academy

Gravity is a conservative field — so as anything falls, energy just trades form. Put a roller-coaster cart in your hands and watch it hold. A coaster cart sits at the top of a hill, height h above the ground. Let it go and it speeds up as it drops — but the books always balance: the height it gives up becomes exactly the speed it gains, and the total never changes. Drop it — energy just changes form Set a starting height, hit Drop, and watch the cart fall. At the top it's all potential energy; at the bottom it's all kinetic . Watch the two bars trade off — while their total stays flat . Every joule of potential energy at the top shows up as kinetic energy at the bottom. Set the mass aside and it's pure bookkeeping: drag the drop height and read off exactly how fast the cart is going when it hits the ground. The path doesn't matter — that's what "conservative" means Here's the deep part. Bend the track between the same start and finish into any wild shape you like — a gentle ramp or a plunging, looping monster. The speed at the bottom comes out identical, because gravity's work depends only on the height dropped, not the route. A force is conservative when it's the gradient of a potential — then energy is stored, never lost, and only the endpoints matter. The same idea runs a planet's orbit ( ), a stretched spring ( ), and the voltage in a circuit. Spot the potential f and you already know the work: .

This is the written version of the interactive lesson above. See the full Calculus 3 course.