The Roller Coaster or Brachistochrone Problem

A roller coaster ride begins with an engine hauling a train of cars up to the top of a steep grade and releasing them. From this point on the train is powered by gravity alone and the ride can be analysed by using the fact that as the train drops in elevation its potential energy is converted into kinetic energy.

It is not too hard to derive a formula for the time t required for a ride along any curve y = (x) that the roller coaster track takes. Click here to see the details of the analysis. The result is the following formula:
Once the curve (x) for the roller coaster track is given, it and its derivative y'(x) can be substituted into this formula and the integration can (hopefully) be carried out. (If the integration can't be done analytically then at least it can be done numerically.)

Brachistochrone is Greek for "shortest time". The brachistochrone problem is to find the curve of the roller coaster's track that will yield the shortest possible time for the ride. This problem was originally posed as a challenge to other mathematicians by John Bernoulli in 1696.

Although we won't prove it, the curve of shortest time is a cycloid, which also happens to be the curve traced out by a point on the rim of a wheel as the wheel rotates. Here is a picture of a cycloid traced this way. The wheel is shown in blue and the cycloid is shown in red. measures the angle through which the wheel has rotated in radians. The parametric equations for the x and y coordinates of the cycloid in terms of are also shown:

If we take one-half of a cycloid and turn it upside-down we get the brachistochrone for the roller coaster:

Notice that it is vertical at the start to get up lots of initial speed and then flattens out at the end. This path has another interesting property, namely if the ride could somehow be started from rest at point B or C, the ride would last the same length of time as if it started from point A.

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