With Only Conservative Forces, Kinetic Plus Potential Energy Stays Constant

Drag the height h slider to lift the ball. The gravitational potential energy is PE = mgh, mass times gravitational acceleration times height. The higher you lift it, the larger the potential energy, in direct proportion. This is exactly equal to the work done against gravity to raise the ball to that height; that is, potential energy is the work gravity will later give back, stored in advance. You only need to fix one reference: where to call height zero is up to you, and what matters is the difference in height. Potential energy is the energy something has from its position alone, even with no motion yet, the potential to fall.

Now drop the ball and watch the two energies together. Lower the ball's height with the slider. The potential-energy bar shrinks, and the kinetic-energy bar grows by just as much. Their sum, the total bar, does not change length throughout the fall. This is the conservation of mechanical energy: KE + PE = constant. Not a bit of the potential energy it had up high disappears; it merely transfers into kinetic energy. So if you know the height and speed at just the start and the end, you can solve it without knowing the in-between. It does not matter whether the path down is straight or curved, a slide or a free fall. But this clean conservation holds only when the forces acting are conservative ones like gravity, whose work is independent of the path.

The most beautiful example is the pendulum. Swing it left and right with the slider. At the far ends, where it rises highest, it pauses for an instant. With zero speed the kinetic energy is zero, and all it has is potential energy. Released from there, as it comes down the potential energy turns into kinetic energy, and it is fastest as it passes the very bottom. There the height is lowest so the potential energy is zero, the moment everything has become kinetic energy. Then, rising up the other side, it gives the kinetic energy back as potential energy. The two bars hand off like a seesaw, but their sum is always the same. With no friction the pendulum rises to exactly the same height every time and swings without end. This exchange is the clearest picture of conservation.

The conservation law hands us one clean formula. The speed an object reaches at the bottom, dropped from rest at height h, is v = √(2gh). It comes from mgh = ½mv², where the mass m cancels from both sides. So a heavy ball or a light ball dropped from the same height arrives at the same speed. Drag the height h and the speed grows as the square root of the height: you must quadruple the height to double the speed. Moreover, this speed is independent of the path taken. Whether it falls straight down or rides a frictionless slide that loops around, as long as the height lost is the same, the arrival speed is the same. This is why a roller coaster's speed at the bottom is set by the height of its first drop.

Then why, in the real world, does a pendulum eventually stop? Because of non-conservative forces like friction and air resistance. Toggle friction on and off with the button. With no friction the total-energy bar stays full, and the pendulum returns to the same height every time. Turn friction on and the total energy drops a little with each pass. The lost energy has not vanished; it has scattered into heat and sound. Energy itself is always conserved, but the mechanical energy we can readily use (kinetic plus potential) shrinks when non-conservative forces are present. So when they act, you cannot say KE + PE is constant; instead, its decrease equals the negative work done by friction. Conservative forces lend energy and give it back, but non-conservative forces let it leak away along the road.