Roller Coaster Physics

Roller Coaster Physics

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Undoubtedly roller coasters are the kings of amusements parks. Whether you enjoy a older, rickety wooden roller coaster with its thrill of positive and negative G's and a fairly wobbly feeling. Or, you prepher something newer, a tube steel roller coaster. A coaster that is faster, one that includes death defying speeds, hairpin turns, and of course the cr�me de la cr�me, loop de loops.

However, regardless of you personal tastes and preferences, through exploring this page you will find that all roller coasters are indeed bound by the same fundamental laws. Laws that govern everything in our daily lives, the laws of physics. While exploring this paper, please remember this simple fact:

Roller coasters are fast, they're fun, they're exciting, but above all, they're PHYSICS!

The basic physics that apply to roller coasters can be seen when we examine some of the simple thrills of roller coasters:

* The relation between Height and speed
* Positive and Negative G's
* The corkscrew
* The loop de loop

Some of you out there might be wondering, what exactly I mean that when I say that there is energy associated with roller coasters? And the answer is very simple, although roller coasters don't produce, or use energy as most people today would define it--electricity. They do posses what physicists call kinetic (or mechanical) energy, which is the energy of motion and is defined with the equation:

K=½mv²
which is read: �Kinetic Energy equals one-half mass times velocity squared.�


However, there is another type of energy associated with roller coasters, and that is gravitational potential energy, which is simply the energy that the roller coaster has due to its position above the earth, and has the formula:
U=mgh
which is read: "Potential Energy equals mass times velocity times height."


Then, when we take into account the First law of thermodynamics (also called the conservation law), seen below:

The First Law Of Thermodynamics:
�Energy can be changed from one form to another, but it can not be created or destroyed.�
click here to see the source page.

So, after taking thermodynamics into account we see that at any given point during the roller coasters ride, (granted we are using a traditional roller coaster in which there are no extra chains, or engines to lift it other than the first hill, and that friction is negligible) we see that:

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Kinitial+Uinitial=Kfinal+Ufinal

Now, with a little algebra, and an initial velocity of 0, that the equation becomes:

v=(2g(hinitial-hfinal)½
or that:"velocity equals the square root of 2 times gravity times the difference in height"


It is because of this correlation between kinetic and potential energy which explains whey roller coasters never exceed the height of their first hill. Notice, that if a height of a hill is equal to the first hill height, hinitial-hfinal=0 and so v=0, or there is no movement, so all hills have to be smaller than the first hill. Also notice, that if the second hill is larger than hinitial-hfinal will be negative, and since we can't take the square root of a negative without using imaginary numbers, this too is not possible.

Friction:
As some of you may have noticed, I simplified my explanation by cutting out friction, a common thing for physicist just concerned with explaining the topic and not practicality. However, in the real world friction is a force that is always acting opposite the direction of motion, and thus we will loose energy and velocity with distance. But friction is a force that, if we know the materials and loads acting on them, can be predicted and compensated for. But for ease and clarity, friction will be set as zero for my entire web page, but remember that it is always there and acts on the system against the direction of motion.

Vairables:
Another thing I have not done is clarify what my variables mean, m=mass and in the metric system is kilograms (kg), and in US customary is the slug. So for ease, from here out when mentioning a new unit, I will define it as like this:
variable=definition (metric unit; US unit).
So I would define mass this way: m=mass (kg; slug), v=velocity ((m/s); (ft/s)), g=gravitational acceleration (9.8(m/s2); 32(ft/s2)), h=height (m; ft).

BE CAREFUL!
It is important to note, that although you can do calculations either in US customary or in metric, you can not mix the two, so, to get f=force (ma; slug(a)), where a=acceleration(m/s², ft/s²) you can not multiply mass in kg by acceleration in ft/s².

What are G's?
One of the most popular attributes of roller coasters is their ability to, in a single ride make the rider feel as though they had doubled their weight, or oppositely, lost all weight. These feelings of either intense weightiness, or weightlessness, are referred to as G's, and are simply referring to how many times of gravity are affecting you. For example, when you go down the large hill at the beginning of a roller coaster, and you reach the bottom of the hill and start climbing the next hill you feel as though you are three times as heavy as you are. Somehow, you find a way to measure your weight at that point in time, and find that it is indeed three times larger. Then, you are experiencing a force of 3G, or three times the force of gravity.

Or if you are going over a sharp hill, you may experience yourself feeling great pressure against your restraints, and this is because of the same phenomena, and are referred to as negative G's because they are acting against gravity.



What Causes G's?
These feelings, as well as some of the other effects of a roller coaster are caused by centripetal force. Centripetal force=Fc=mac, ac=centripetal acceleration=m(v2/R), where R is the radius of the circle. So that:
Fc=m(v2/R)

Notice, that because v is squared, then if you double your speed with all other variables constant, you actually quadruple (22=4) your centripetal force. So if you originally felt 1G of force on a hill, you will now, at twice the speed experience 4G's. Similarly, if you triple the speed then you will have NINE times the force.



This raises many concerns for roller coaster engineers, because the average human body can withstand around only 5G's for only a few seconds before blacking out, a condition where either too much blood or not enough blood is circulating to the brain. For this reason, most roller coasters with grater speed have significantly larger radii on their flips and turns. But remember that if roller coaster "A" has a speed that is two times roller coaster "B", to keep the same G's "B", roller coaster "A" must have a radius 4 times that of "B".

However, centripetal force is not only a vertical thing, in fact I would wager that almost all of those reading this have themselves experienced centripetal force, regardless of whether they rode a roller coaster or not. How? Well in your car, any time you take a tight turn and feel your self being pulled to one side or the other you are experiencing centripetal force.

What is a corkscrew?
A corkscrew is a series of small consecutive circles, or clothoid circles (slightly oblong circles), that will give the rider a combined feeling of G's and disorientation.



Corkscrews are also made possible through centripetal force, because as the cart is being carried through the rotations it is being pushed outwards towards the structure. However, if there is not enough velocity to carry the passengers all the way through the corkscrew than the forces needed to keep the car pressed up against the track will not be significant enough to overcome the weight and the cart would be liable to fall if it weren't for the unique wheels that hold the cart to the track.


A loop de loop is simply a loop that the roller coaster makes. Although a loop de loop is very similar to a corkscrew, it has its own unique physics. Like I mentioned some corkscrew will have clothoid loops, but the vast majority of them are indeed circles. However, loop de loops, because of their typically superfluous height will almost always be clothoidal.

So what is so special about a clothoid?
Well, a clothoid is evident proof that designers of roller coasters must always be paying attention to the physics or their job. As I mentioned in the corkscrew section, a clothoid is a slightly oblong circle, and has a sorta tear-shape to it. But what does that do? Well, this shape makes it so that the loop has a smaller diameter at the top than it does at the bottom, and this is significant because it allows the coaster to keep moving with roughly the same acceleration while it does the loop. Remember that the centripetal acceleration is v2/R, so by decreasing R they can increase the acceleration of the cart.

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