Additional notes: GPE = m x g x h KE = m x v₂
The main energy transfers that happen as a car travels along the track from the start of the ride to the end:
The rollercoaster car gains gravitational potential energy (GPE) as it travels to the top. Once over the top, the car gains speed as GPE is transferred to kinetic energy (KE). As it travels to the top of another loop, KE is transferred to GPE. Not all the energy is transferred to or from GPE – some is transferred to the surroundings as heat and sound. All moving objects have kinetic energy, KE. The kinetic energy an object has depends on the mass and speed. If the mass doubles, the KE doubles and if the speed doubles, the KE quadruples. Normally energy is lost through sound and heat (friction, air resistance).
How the heights of the hills are designed to allow an empty car to reach the end of the ride:
The hills are designed so that it is low enough that the momentum of the car from the previous drop carries it up and over the hill. This is why the hills are usually lower towards the end of the ride, because the car has lost momentum due to friction and air resistance. Mainly the consecutive hill must be lower as it will not have enough energy because some of it is lost and sound and heat. Therefore, if the car was to reach the end of the ride, the height of the hills must be lower each consecutive time.
How the energy transfers determine the heights of the hills:
 After the roller coaster is drops from the first hill it does two things with its energy. First, it begins to transform that energy from one form to another--from gravitational potential energy to kinetic energy and from kinetic energy to gravitational potential energy, back and forth. Second, it begins to transfer some of its energy to its environment, mostly in the form of heat and sound. Each time the roller coaster goes downhill, its gravitational potential energy decreases and its...