Re: What shapes do roller coaster hills/dips follow? (parabolas? circles? etc.)

Newer roller coasters are designed by computer, and the designer can throw
in all sorts of twists and whistles in order to provide an exciting ride.
I'm assuming that you are designing a more conventional coaster, though.
These rides usually use a segment of a circle as the horizontal curve, 
simply because
it is easier to compute.  In order to keep the riders more or less in 
their seats, the tracks are superelevated, or tilted
in towards the center of the circle. 
 http://www.vast.org/vip/book/BANKEDCU/HOME.HTM http://www2.glenbrook.k12.il.us:591/coaster/designdirns.html
 
In order to get from the straight track to the curved, superelevated
track, a track segment called a cubic spiral is sometimes used.  This is a
mathematical construct that increases the curve of the track as the
superelevation of the track increases, so that the centrifugal forces are
balanced.  Thus, a string of roller coaster cars going down a track would
encounter a straight segment; a spiral segment where the radius and
superelevation increase; a circular segment of constant radius; another
spiral, this one in the opposite direction; and another straightaway.
The mathematics for calculating a spiral are fairly daunting, and these
sections of track are short.  So for what you are doing, I'd just go with
segments of circles.  A 'coaster designer wants the riders to enjoy an
exciting ride anyway, so she will deliberately shorten up the spirals and
flatten the superelevation so that the riders get slammed around, within 
the limits of safety.
Because a curve will have greater energy losses due to the friction of the
wheels rubbing on the outer rail, the curved track must either go downhill
slightly, or the speed will decrease as the cars go through the curve.  You
will have to make allowance for this, or your cars are liable to come to a
stop after a string of curves!

Vertical curves are usually a parabola.  The idea here is to reduce the
force of gravity on the rider.  Some vertical curves go so far as to make
the rider feel weightless!  
This is much simpler mathematically than a spiral.  Let's say that your
coaster approaches a summit at an angle of 25.26 degrees, and a speed of 67
feet per second.  If you lay this out, you'll see that this rather oddball
set of numbers has the coaster cars traveling horizontally at 64 feet per
second, and upwards at 32 feet per second.  Now, if we want the car and
riders to feel weightless, we must design the curve so that they can
accelerate downwards at 32 feet per second.  The formula is 
V(t) = V(0) + at. V(0) is 32, a is the constant of acceleration by 
gravity, which is 32 feet per second, squared.  Solving this for t means 
that in this case, one second after we start, the velocity upwards is 
zero, i.e., the car has reached the top of the hill.  In that same one 
second, the car travels horizontally 64 feet.  So you'll need a hill 2 X 
64 or 128 feet long to get two seconds of weightlessness.  Needless to 
say, these numbers (except for a) can be changed to get different 
results.  Reducing the weight of the riders by half or more will probably 
make them think they're about to fly out of their seats, even when they're 
not.
The track height above this initial point is given by the formula   S =
V(0)T + ½ aT(squared), with a being negative.  So, to calculate the height
of the crest of the hill (one second of travel, remember), it is 32 x one
second, minus ½ (32) x one squared, or 32 - 16, or 16 feet above the 
initial
point.  You can break the time down into shorter intervals, calculate the
track elevation and plot the curve.  The shape will be a parabola defined 
by
gravitational acceleration. 
(Hey, there's a stunt for you - design the first roller coaster on the 
Moon,
where the acceleration constant is 1/6 what it is here!)

Here are the sites of a couple of companies that design roller coasters:
 http://www.pribonic.com/ http://www.arrowdynamics.com/

And some more Links: http://www.kyrene.k12.az.us/itech/amsitech/activities/coaster/links.html