Physics Behind Roller Coaster Waves

History
A roller coaster train going down hill represents merely a complex case as a body is descending an inclined plane. Newton's first two laws relate force and acceleration, which are key concepts in roller coaster physics. At amusement parks, Newton's laws can be applied to every ride. These rides range from 'The Swings' to The 'Hammer'. Newton was also one of the developers of calculus which is essential to analyzing falling bodies constrained on more complex paths than inclined planes. A roller coaster rider is in an gravitational field except with the Principle of Equivalence.

Potential Energy
Potential energy is the same as stored energy. The "stored" energy is held within the gravitational field. When you lift a heavy object you exert energy which later will become kinetic energy when the object is dropped. A lift motor from a roller coaster exerts potential energy when lifting the train to the top of the hill. The higher the train is lifted by the motor the more potential energy is produced; thus, forming a greater amount if kinetic energy when the train is dropped. At the top of the hills the train has a huge amount of potential energy, but it has very little kinetic energy.
Kinetic Energy
The word "kinetic" is derived from the Greek word meaning to move, and the word "energy" is the ability to move.
Thus, "kinetic energy" is the energy of motion --it's ability to do work. The faster the body moves the more kinetic energy is produced. The greater the mass and speed of an object the more kinetic energy there will be. As the train accelerates down the hill the potential energy is converted into kinetic energy. There is very little potential energy at the bottom of the hill, but there is a great amount of kinetic
energy.
Theory
When the train is at the top and bottom of the hill there is not any potential or kinetic energy being used at all. The train at the bottom of the first drop should have enough energy to get back up the height of the lift hill. The "Act of Faith" in riding these amazing rides which seems more of a phenomena that is only a theory. In practices, the train never could make it back up the hill because of dissipative forces. Friction and air resistance, and even possible mid-course breaks, are dissipative forces causing the theory to be changed but not destroyed. These forces make it impossible for the train to have enough energy to make it back up the lift hill's height. In the absence of the dissipative forces the potential and kinetic energies(mechanical energy) will remain the same. Since the mechanical energy is destroyed by the forces, the first hill is always the highest.