conservation of mechanical energy explained
Lift a ball high and let it go. Once it leaves your hand it falls faster and faster, but even before it moves, a raised ball holds a readiness to fall: potential energy. It is the work gravity will do, stored in advance. As the ball falls, that potential energy turns into kinetic energy, and the ball gains speed in proportion to the height it loses. The remarkable part is that, as long as there are no nonconservative forces such as friction or air resistance, the sum of the two never changes during the fall. Energy only moves between motion and position, the total stays the same, and that is the conservation of mechanical energy. It is a shortcut that solves motion with a single energy ledger instead of tracking force after force.
Lift the gold ball with the height slider h and the gravitational potential energy in the green bar, PE = mgh, mass times gravitational acceleration times height, grows in direct proportion. This value is exactly the work done against gravity in lifting the ball to that height, so potential energy is work stored for gravity to give back later. You only have to choose one reference: where you put height 0 is up to you, and what matters is the difference in height. Potential energy is energy held by position alone, before any motion, a potential to fall.
Now drop the ball and watch both energies together. Lower the ball with the slider and the green potential-energy segment shrinks while the blue kinetic-energy segment grows by the same amount; the total length of the bar stays the same throughout the fall. That is conservation of mechanical energy, KE + PE = constant. The potential energy the ball had up high does not lose a single joule; it simply moves into kinetic energy, so knowing the heights and speeds at the start and end is enough to solve a problem without knowing what happens in between. It makes no difference whether the path is straight or curved, a slide or a free fall. This tidy conservation holds, however, only when the only forces doing work are conservative ones, like gravity, whose work does not depend on the path.
The most beautiful example is the pendulum. Tilt it to either side with the slider: at the two highest points the speed is 0, so the kinetic energy is 0 and all the energy is potential. Coming down, potential energy turns into kinetic energy, and at the lowest point the potential energy is 0: everything has become kinetic energy, so this is where the bob is fastest. Climbing the other side, it turns kinetic energy back into potential energy. The green and blue segments trade places like a seesaw while the total bar length never changes, and without friction the pendulum rises to the same height every time and swings forever. This exchange is the clearest picture of conservation there is.
Conservation hands us a tidy formula as a bonus. An object released from rest at height h reaches the ground with speed v = √(2gh), because in mgh = ½mv² the mass m cancels from both sides. So a heavy ball and a light ball dropped from the same height arrive at the same speed. Drag the height slider h and the arrival speed shown by the blue arrow grows as the square root of the height: you need four times the height to double the speed. The speed does not depend on the path either: falling straight down or winding down a frictionless slide, the same loss of height gives the same arrival speed. That is why the speed at the bottom of a roller coaster is set by the height of its first drop.
Why, then, does a rolling ball eventually stop in real life? Because of nonconservative forces such as friction and air resistance. The figure shows how high a ball released at the left rim of a bowl-shaped track climbs on the far side, and the buttons switch friction on and off. Without friction the ball climbs back to the dashed starting height and the whole bar is green mechanical energy. With friction it stops at the lower red dashed height, and a red heat segment of matching size appears at the end of the bar. The missing energy has not vanished but scattered as heat and sound, so although energy as a whole is conserved, the mechanical energy we can use (kinetic plus potential) decreases. With nonconservative forces present, then, KE + PE is not constant; instead the decrease equals the negative work done by friction.