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Name: Joe F.
Status: student
Age: 16
Location: N/A
Country: N/A
Date: 2001-2002

Hi, I am doing a roller caster project in my physics class. Every thing I've looked up on the Internet about coaster physics begins with "Neglecting air resistance...". What if I did not want to neglect air resistance what if I wanted to calculate it in my equations. How would I do that?


Including air resistance means a great deal more work. Air resistance takes into account the work necessary to move the air molecules out of the way so the cart can move forward. There is no specific formula for air resistance. It depends on the shape of the roller coaster cart: a flat front results in more air resistance than a rocket-shaped front. It depends on speed of the roller coaster, which is constantly changing. It depends on temperature and humidity, how dense the air is at that particular moment. If a scientist wanted to include air resistance, he would calculate the motion without air resistance for a similar roller coaster, make actual measurements for that similar coaster, and then figure an average effect of air resistance for that kind of roller coaster.

If you want the formula for air resistance, sometimes called aerodynamic drag, a formula that works well is (drag force)=-b*(speed)^2. Aerodynamic drag is proportional to the square of the object's speed. The "b" is a constant determined by experiment, based on all things mentioned above. There is no theoretical value for "b".

If the cart of a roller coaster is aerodynamically designed (not just a flat plate on the front) and has a significant amount of mass (maybe 500 lbs, with riders included), air resistance will not have a huge effect.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

Neglecting air resistance is a pretty safe assumption on most coasters since the only surface area that would contribute to air resistance is the front part of the roller coaster. The coaster is so massive relative to the small area at the front contributing the most to wind load. Of course, this becomes less true for coasters like Batman at Magic Mountain, where your legs are hanging out in the air. Coasters like this would probably have a greater wind resistance / length of coaster. To qualitatively account for this you would need to consider the front part of the coaster as a wind load as well as the many tentacles (peoples legs) hanging from the large mass. To quantitatively account for the drag from peoples heads and legs would require a little knowledge of fluid dynamics and Stoke's Theorem. It is a pretty safe assumption to neglect air resistance.

-Darin Wagner

If you do not want to neglect air resistance, then you have a problem.

Air drag depends on the air density, the velocity squared, the cross-sectional area of the object, and the air-drag properties of the object.

Fine, that seems easy enough, except when you start trying to put numbers to these things and trying to solve them. A simple equation for drag is:

D = 1/2 * C * p * A * v * v

Drag equals on half of the drag coefficient times the air density times the cross-sectional area times the velocity squared.

So, when trying to calculate the velocity of the coaster, you will find that the velocity will depend on the acceleration, but the acceleration depends on the force, which in turn depends on the velocity squared. This is a differential equation, and it cannot be solved using simple algebra, or even advanced algebra. Additionally, the velocity will also depend on the wind speed in the direction of the coaster travel, and this will change throughout the ride.

Next, you have the density of air. This depends on the temperature, altitude, and humidity. You can use an average, but then you are not going to get an exact result.

The cross-sectional area is next. For a coaster, this will be a big problem to calculate. Typically, they are not simple circles, squares, etc. so it will have to be measured in some fashion. Additionally, the area will change during the ride. As you go around a corner, for example, different parts of the train will be moving in different directions. You always want the area in the direction of travel. So for each point along the track you will need to figure out the area. Again you could simply use one area and hope for the best. Additionally, the area will depend on the shapes and sizes of the riders, and will thus change for each ride.

Finally, you will need the drag coefficient. The way you get this is by measuring it in a wind tunnel or some other such mechanism. There really is not any way that it can be accurately determined any other way. The type of paint, exact shape, and all kinds of other things will affect this number. Again, this will change as you go around curves, and the riders change.

So once you have all this figured out for every point along the track, you are ready to rock and roll. However, you will have had to make many simplifications, and your numbers will not be very accurate, probably not even accurate within 50%. Additionally the equations will be extremely complicated. But if you do all that work, you will find that the air drag contributes a small amount to the overall forces. Coasters move at relatively slow speeds (compared to their terminal velocity), and so the velocity squared factor makes the air drag diminish rapidly at relatively slow speeds.

So, most people do the sane thing, realize that either way they will have to fudge the answers, so they say "Neglecting air resistance..." and use the simpler set of equations.

Alternatively, some computer systems model the air as millions of tiny particles, and simply run a simulation of the whole thing. If you have a software package that can do it, that is your best bet. The way this works is that at a given moment all the forces are calculated and determined, everything is moved a little bit, collisions are detected and resolved, and all the new velocities are worked out. Typically a few hundredths of a second are simulated each step. The math is quite simple, and it works fairly well. It again is not completely accurate, but can be made more accurate by taking smaller time slices and adding more particles to the simulation.

Hope this helps.

Eric Tolman
Computer Scientist

Roller coaster friction is significant. It is also remarkably complex and has no single, easy, formula for application. There is bearing friction which is temperature dependent. The air resistance varies by cross-sectional profile of the car, particularly the front car. Track conditions, materials, and wheel materials have influence. The barometric pressure, humidity, winds, and other atmospheric conditions effect the friction. Velocity profile of the ride matters, since air drag is not a linear function. Roller coasters used to be designed with a "friction profile" that was done more "by the seat of the pants" rather than calculated. We now have computer modeling, where every few milliseconds, we can model what is happening. It is an enormous amount of calculation. The different design companies closely guard this as proprietary information.

---Nathan A. Unterman
Nathan A. Unterman
Glenbrook North High School

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