**Question** – I am doing a 7th grade science fair project on the physics of roller coasters. Can you give me some useful information about this?

As you have probably learned, roller coasters operate by exchanging two types of energy: “potential energy” and “kinetic energy”. Potential energy is the result of the height of an object, the height of the roller coaster vehicle in this case. Kinetic energy is the energy of motion, the motion of the roller coaster vehicle.

Typically, a motor operating a chain does “work” to lift the roller coaster vehicle to the top of the highest hill, giving the vehicle its maximum potential energy. As the roller coaster falls down the first hill the vehicle’s potential energy converts to kinetic energy as the speed of the roller coaster vehicle increases. Then, the vehicle surrenders its kinetic energy as it slows down to climb the next hill, regaining some of its original potential energy. The vehicle goes along trading potential energy for kinetic energy, hill by hill. It is always losing a little bit of its energy to friction. Because of this, each hill must be smaller than the one before, until eventually all of the original potential energy has been consumed by friction. At this point it is not possible to go over any more hills, but only to coast back into the station. If it were possible to completely eliminate friction, a roller coaster vehicle could go around and around the track, forever trading speed for height and vice versa.

We can apply the basic laws of physics to answer questions like “how fast will the vehicle be moving at any particular point on the track” and “how high can the vehicle climb before it runs out of speed?” To answer these questions, all we need to know is the speed and height of the vehicle at its present position, and the height of the future position. The difference between the two heights tells us the change in potential energy. Potential energy changes equal kinetic energy changes, so the change in height tells us the change in speed. Similarly, the change in speed tells us the change in height.

It is almost correct to say that the weight of the vehicle (the number of riders) does not matter. But, because of the problem of friction, the weight of the vehicle does matter a little bit. We can leave this complexity for another day.

If you do not mind a little math, it will help to know that the formula for kinetic energy is

E_{k} = ½ mv^{2}

where m is the mass and v is the velocity. (Mass is essentially the same thing as weight.)

The formula for potential energy is

E_{p} = mgh

where m is the mass, g is acceleration due to gravity (9.8 m/s^{2}) and h is the height.

With the help of a little bit of algebra you can solve for velocity by setting the two forms of energy equal. You will notice that mass disappears from the equation – it doesn’t matter. You will find that the increase in the speed of the vehicle is

v = (2gh)^{½} = 4.4 h^{½}.

So, if we ignore the small changes due to friction, the change in velocity is a direct result of the change in height, and vice versa!

From this formula you can see that if a roller coaster vehicle starts out at the top of the hill without any speed, by the time it falls just 5 meters (about 16 feet) it will have reached a speed of about 10 meters/second (about 22 miles/hour). It doesn’t even matter how steep is the hill.

These are some useful websites about the physics of roller coasters:

http://www.sciencebuddies.org/science-fair-projects/project_ideas/Phys_p037.shtml

http://search.eb.com/coasters/ride.html

http://www.howstuffworks.com/roller-coaster.htm

http://cec.chebucto.org/Co-Phys.html

http://www.worsleyschool.net/science/files/roller/coasters.html

** Disclaimer:** These are the thoughts of the author and do not represent an engineering or legal opinion of Birket Engineering, Inc.