According to a recent article in *Popular Mechanics*, the construction of
roller coasters with loops is responsible for a resurgence of
popularity for these amusements; loops are also implicated in several
deaths. In this project, you will investigate some of the crucial
aspects of roller coaster design.

One of the first companies to produce a coaster with a loop was Schwarzkopf GmbH of Germany. The so-called Schwarzkopf loop is described in the article as follows.

Sixty-three-year-old former cartwright Anton Schwarzkopf holds nearly 60 patents in the field of roller coaster design and manufacturing, and has produced more than 55 roller coaster systems around the world. His loop is the result of rigorous research and development conducted in the early '70s at his Bavarian test track. Other roller coaster makers had designed loops, but their ``geometry'' had imposed too many Gs on passengers, resulting in whiplash, broken collar bones, bruises , and other bodily strains. Schwarzkopf determined that a safe loop consisted of a spiral in which the radius of curvature decreased at a constant rate. Thus, most modern loops are tear-drop or oval-shaped, which means riders are subjected to slightly less than 6 Gs as the roller coaster's cars enter and leave the loop.

The key phrase in the passage above is *...radius of curvature
decreased at a constant rate*, at least as far as this project is
concerned. In general terms, your first task is to find out what
radius of curvature means and why it is important in coaster design.
The second task is to investigate the Schwarzkopf loop. Can you come up with a
model that retains the essential features, but is simple enough that
you can solve it?

To do this project, you will have to read and understand the material in §14.4 and 14.5 in the text. This is where the unit tangent and normal vectors and of a curve are described, as well as the notion of a the curvature, of a curve.

In designing a roller coaster, the most important concepts are speed and force. You want the speed to be high, but not too high and the same goes for the forces on the passengers. According to the article quoted above, the total acceleration should not exceed 6 Gs. Obviously, we don't have a test track in Bavaria, so we will have to construct a simplified mathematical model. Here is the first set of assumptions.

- A track can be represented by a curve in three dimensions parametrized by
**s**. Here**s**is arc-length along the curve. - A roller coaster car can be represented by a particle moving along the curve . This assumption means that we won't be able to account for internal forces on the passengers or for torques caused by the car having a finite height. These are certainly important aspects of roller coaster design, but they are way beyond the scope of what we are trying to do.
- The actual motion of the car, , on the track can be
represented by a ``speed function'' , so that we have
- Friction between the car and the track will be neglected. This means that the tangential force exerted on the car by the track will be zero.
- The only forces on the car will be the normal force, and gravity .

Under these assumptions, Newton's second law and the decomposition

can be used to obtain the two scalar equations

Equation is a differential equation that will have to be solved for the speed function . Equation allows one to compute the normal acceleration, if you know the curvature and the speed . It wasn't made explicit in the equations, but , , , and all depend on time because the car is moving along the curve and , , and all change as the car moves.

Here are some specific things I want to see in your report, as part of the usual Introduction, Background, Procedure, and Conclusion structure. The next section of this handout provides hints and clues that should aid your investigations. You also might want to do some research in the library, including past MQPs.

- Details of how Equations and were derived.
- Your own version of the ``Schwarzkopf loop''. Provide plots of
speed, total acceleration, and normal acceleration for your design. Any
additional assumptions you have to make to get results must be clearly
explained and justified. You should also describe how you obtained
your loop and details of how , , and vary
along your loop.
- An analysis of your model's shortcomings and suggestions for improvement.

Here are some suggested activities, designed to help you construct and analyze your own ``Schwarzkopf loop''.

- Choose some simple curves, e.g. a straight line or a circle, and see if you can solve Equation for and evaluate in these special cases. For a straight line, the answer should be yes. For a circle, look up the simple pendulum equation in your Physics book.
- What if you were designing a roller coaster for a weightless environment ? How would that simplify your computations?
- The ``Schwarzkopf loop'' is described as having a radius of
curvature that decreases at a constant rate along the track. What does
this imply about as a function of
**s**? - A related problem is the following. Given a curvature, , can you find a plane curve that has this
curvature? The answer is yes, and I suggest you study the curve
defined by
where, hopefully, the function can be related to the curvature in some simple fashion. Try to generalize to a curve in three dimensions.

- The curve in the previous item had constant speed. How would you modify it to get a variable speed?
- If you can't solve Equation analytically, you might
investigate Maple's
`dsolve`procedure, especially with the*numeric*option. The*Maple V Flight Manual*(I have a copy) might also be useful.

This is an open-ended project. That means, for one thing that there is no unique ``right'' answer. Here is what I think is a reasonable schedule for the completion of this project.

- September 18 (!). Problem statement handed out.
- September 25. Progress reports due.
- October 2. Draft reports due. They will be returned with
comments on Monday, October 5.
- October 9. Final project reports due.

Wed Jul 26 13:43:32 EDT 1995