Re: When Physical Intuition Fails

[John Clement <clement@HAL-PC.ORG>]

> > Incidentally the UMass Amherst PER group had a similar problem that
> > stumped many experts and students alike.  It was a car rolling down
> > a slight incline vs a car rolling into a hole and then to the same
> > final location.  It was surprising how many were surprised to see
> > that the longer path yielded a shorter time.  They showed the result
> > with movies and simulations.
>
> There is a very good chance that very many on this list know about
> the brachistochrone problem since it has been around for literally
> hundreds of years.  Galileo showed by explicit demonstration
> (remember, he loved to roll things down ramps) that the cycloid-
> shaped path is the minimal time path.  The problem was solved
> analytically by J. Bernoulli over 300 years ago and is a classic
> (probably *the* classic) example of the application of the
> calculus of variations.
>
> A point to keep in mind about this problem is that the over all shape
> of the minimal time path depends on the ratio of the vertical height
> of descent between the endpoints to the horizontal range between the
> endpoints because this ratio determines the total parameteric
> rotation angle of the cycloid.  For instance, if the net height of
> descent is greater than 2/[pi] times the horizonal range then the
> minimal time path does *not* dip below the level of the final
> endpoint and there is no dip or "hole".

Actually it is a little different because it uses a low friction cart.  As a
result there is no rotational kinetic energy to consider and the problem is
actually simpler.  The problem was used as a probe of student understanding
of dynamics ideas.  To show that the time is shorter does not require
complex math considerations, and can be figured by some simple arguments.
One of the interesting points was that students realized that you could have
a straight path in the hole on which the speed is higher than on the
original path.  Most students then incorrectly reasoned that the straight
path was necessary to make the effect work.  As I recall it also stumped
some "experts", but I do not remember how many.  Galileo didn't have low
friction carts, but I suspect he would have loved to use them.  There are so
many "classic" problems that it is impossible to know them all.  Pasco has
since created a cute line of small low friction carts with flexible tracks
for this problem.

John M. Clement
Houston, TX

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.