Energy conservation in roller coasters illustrates the transformation between potential energy and kinetic energy, and vice versa.

Roller coasters serve as an excellent demonstration of the principle of conservation of energy, which asserts that energy cannot be created or destroyed; it can only be transferred or converted from one form to another. In roller coasters, the two primary forms of energy at play are potential energy (PE) and kinetic energy (KE).

At the beginning of the ride, the roller coaster car is lifted to the top of the first hill—the highest point on the track—using a chain lift or other mechanical systems. At this peak position, the car possesses maximum potential energy, as it is situated at the greatest height. Potential energy is defined as the energy an object has due to its position within a gravitational field, and it can be calculated using the formula:

where $m$ represents mass, $g$ is the acceleration due to gravity, and $h$ is the height.

As the car descends from the first hill, this potential energy is transformed into kinetic energy, which is the energy an object possesses due to its motion. The kinetic energy increases as the car accelerates, reaching its maximum at the bottom of the hill. The kinetic energy can be expressed by the equation:

where $m$ is mass and $v$ is velocity.

When the car climbs the next hill, kinetic energy is converted back into potential energy as its speed decreases while its height increases. This cycle of energy transformation continues throughout the ride. The total energy within the system (comprising the car and the Earth) remains constant, provided that no energy is lost to friction or air resistance.

In practical scenarios, however, some energy is invariably lost due to friction and air resistance. This is the reason why the first hill is always the tallest. The energy lost to these forces is insufficient to allow the car to reach a height greater than that of the first hill. Consequently, each subsequent hill must be lower than the preceding one; otherwise, the car will not have sufficient energy to ascend.

In conclusion, the conservation of energy in roller coasters involves the ongoing transformation of potential energy into kinetic energy and back again, with some energy inevitably dissipated due to friction and air resistance.