# Conservation of Energy

Conservation of energy means that the total energy within an isolated system will remain constant and will be conserved over time.  This is based off of the fact that energy, like mass, cannot be created or destroyed, but instead can only be transformed into a different energy form.  A perfect example of this is on a roller coaster.  As the train of the roller coaster is being pulled up its first hill it is storing potential energy.  Once the train reaches the crest of the first hill its potential energy is then converted into kinetic energy as it moves down the first hill.  Note the resulting kinetic energy cannot be greater than the potential energy that was converted into kinetic energy.  After the train has transformed the stored potential energy into kinetic energy it will than convert that kinetic energy back into potential energy as it goes up the next hill on the track.  This process will continue to repeat until the end of the ride.

Now you might be saying to yourself that it is common knowledge that the hills after the first hill have to be shorter than the first hill because the train will not return its original starting height, and as the ride progresses it will continue to loose energy.  This energy loss is due to friction from the trains contact with the track, and drag due to air resistance.  Due to this fact the resulting conservation of energy equations for a roller coaster would be the following.

(Eq 1) $U_{1-2}=T+Q$

where

$U$=Potential Energy

$T$= Kinetic Energy

$Q$ = Energy transformed into heat due to Friction, Drag, and braking at the end of the ride.

and

(Eq 2) $U_{1-2}=T_2-T_1+Q$

Equation 1 represents the conservation of energy equation for train on top of the first hill while equation 2 represent the conservation of energy equation for the train for the remainder of the ride.  The reason why there are two kinetic energy variables in equation two is because normally the train does not come to a complete stop as it crests the remaining hills meaning that not all of the kinetic energy is transformed into potential energy. Also, at the end of the ride the value for Q will equal the total potential energy of the train that was stored at the top of the hill since energy cannot be created or destroyed but instead needs to transform into another form of energy.

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