*Conservation of Energy*

Energy cannot be created or destroyed, but it can be converted from one form to another.  For the idealized roller coaster, all energy is conserved through conservative forces, such as gravity.  As the train accelerates down the lift hill, potential energy is converted into kinetic energy.  When the train ascends another hill, the kinetic energy is converted into potential energy again.  This is conservation of mechanical energy, and it continues throughout the entire ride.  The total mechanical energy for the train is shown by the equation:

E = K + U

Where E is the total mechanical energy, K is kinetic energy, and U is potential energy.  From this, the equation for conservation of total mechanical energy can be derived:

E_i = E_f

or

K_i + U_i = K_f  + U_f

Where E_i is total initial mechanical energy and E_f is total final mechanical energy.  This shows that the total initial mechanical energy equals the total final mechanical energy for the system.  It is because of this phenomenon that a roller coaster is called a “coaster.”  After the initial input of energy to carry the train up the lift hill, the roller coaster simply coasts through the rest of the ride.