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 = 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 y (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 onehalf of a cycloid and turn it upsidedown 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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