# Conservation Of Energy Roller Coaster

## Introduction

The Conservation Of Energy Roller Coaster animation on this page explores the conservation of kinetic and potential energy. You can interact with the animation, and immediately see the effects on the roller coaster train. The animation is accompanied by a discussion.

## Demonstration

A roller coaster train is carried uphill on the roller coaster lift hill, which gives the train a huge initial potential energy (height) and extremely small kinetic energy (speed). The train is released and then coasts through the rest of the track, giving up potential energy and gaining kinetic energy when traveling downhill, and giving up kinetic energy and gaining potential energy when traveling uphill.

In this animation, you can...

- Change the shape of the roller coaster by clicking on the numbered gray handles and dragging them.
- Pause the animation by clicking in the Pause checkbox.
- Allow energy losses by clicking in the Energy Losses checkbox. Energy is no longer conserved when this is selected.
- Select which curves are plotted, and how they are plotted, using the dropdown list and checkboxes at the bottom.

The energy curves can be "stacked" or "unstacked". When the energy curves are stacked, the kinetic energy curve is added to the potential energy curve, which helps to show how their total stays constant.

## Discussion

### Potential Energy

There are actually multiple types of potential energy, so it should be stated that the potential energy mentioned in this web page is "gravitational potential energy", which is the energy due to the train's height in a gravitational field. The gravitational potential energy is computed using PE = mass * gravity * height. When height increases, the potential energy also increases.

### Kinetic Energy

There are actually multiple types of kinetic energy, so it should be stated that the kinetic energy mentioned in this web page is "linear kinetic energy", which is the energy described by the train's speed. The linear kinetic energy is computed using KE = 1/2 * mass * speed2. When speed increases, the kinetic energy also increases.

### Conservation Of Energy

Although the potential energy and kinetic energy change quite a bit, their total ideally stays constant (at 100%) starting at the top of the lift hill and while the train coasts around the roller coaster. In this ideal case, we say that the total energy is "conserved", and we can use the following equations:

TE_{a} = TE_{b}

KE_{a} + PE_{a} = KE_{b} + PE_{b}

where TE=train's total energy, KE=train's kinetic energy, PE=train's potential energy, and a and b are any two points between the top of the lift hill and the end of the roller coaster. In this simulation it is often useful to consider the top of the lift hill as point a.

In the real world, energy gets lost in various ways to the surrounding environment over time, so the total energy of the train slowly drops from its initial value of 100%. These unwanted energy losses reduce the train's kinetic energy, and the speed in turn. Energy is still conserved if we consider the train and the environment together, because the energy lost by the train equals the energy gained by the environment. In this real world case, we can use the following equations:

TE_{a} = TE_{b} + EL_{b}

KE_{a} + PE_{a} = KE_{b} + PE_{b} + EL_{b}

where EL=energy lost to the environment.

Of course, there is one energy loss that we want - brakes that slow the train down at the end of the ride so we can get off.

## Discussion Questions

Here are some questions that are answered in the text of this web page, including the equations, or by experimenting with the animation:

- Is the train's energy conserved between the top of the lift hill and the end of the track?
- Is the train's energy conserved while the train is going up the lift hill?
- Where does the roller coaster train go fastest?
- Where does the roller coaster train go slowest?
- Can the roller coaster train ever go higher than the top of the lift hill?
- What is the highest possible value for the potential energy? The lowest possible value? Why?
- What is the highest possible value for the kinetic energy? The lowest possible value? Why?
- Do energy losses to the environment affect the potential energy? How?
- Do energy losses to the environment affect the kinetic energy? How?

Here is a question that is not answered in this web page:

- What are some ways in which the roller coaster train can lose energy to the surrounding environment?