# Conservation of Energy

If you were to look at how energy acts in a system you would notice that the energy can change states.  An example would be compressing a spring.  When you compress the spring you are storing potential energy.  In contrast when you release the spring the stored energy will become kinetic energy.  This is due to conservation of energy.  In other words,  you cannot created energy nor can you destroy it.  Instead you can only make energy change its form.

### Conservation of Energy:  Potential and Kinetic Energy

From the above statement you can see that potential and kinetic energy are opposites of each other.  As a result kinetic energy can be stored as potential energy and potential energy can be converted into kinetic energy.  A perfect real life example of this would be a roller coaster. First, potential energy will be stored as the train is pulled up the first hill.  Once the train crests, the potential energy will be converted into kinetic energy.  As a result the train will increase its speed.  The speed will reach its maximum at the bottom of the hill.  This process will 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.  There are two kinetic energy variable because the train doesn’t come to a complete stop when it crests the top of the remaining hills.  Due to this not all of the trains kinetic energy will be converted into potential energy.  Finally, at the end of the ride, all of the energy must be converted into heat.  This in turn will allow the train to come to a complete stop.

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