Kinetic and Potential Energy

A roller coaster car at the top of the first hill has almost no speed but a lot of height. That height is gravitational potential energy, energy stored because gravity could pull the car down. As the car drops, gravity does work on it, height disappears, and speed builds. At the bottom, all that stored energy has become kinetic energy, energy of motion. That's why the car is fastest at the bottom of the very first hill: it has given up the most height it will ever give up in one drop, so it has gained the most speed.

On a frictionless track, total mechanical energy never changes. A bar chart of kinetic plus potential energy at any point always reaches the same total height, even though the split between the two bars changes constantly. At the top of a hill, the potential bar is tall and the kinetic bar is short. Partway down, they're more even. At the bottom, potential is zero and kinetic is everything. Climbing a second hill just reverses the process: kinetic energy converts back into potential as the car rises, and if the second hill is lower than the first, there's leftover kinetic energy at its top.

This connects directly to the worked problems: setting mgh equal to (1/2)mv² and solving for v is just this conservation idea written as an equation, and it's also why mass cancels out of the speed. It also sets up why potential energy needs a reference point, since 'height' only means something once you pick a zero level to measure from.