We saw that when an object
rolls down an inclined plane, its final speed depends only on the height through
which it descended. Properly speaking, it depends on the "number"
of energy surfaces crossed, but near the Earth, this amounts in practice to
the same thing, because these surfaces are essentially flat and equally spaced
in small regions.
Now there is no reason to
restrict our investigations to flat surfaces like inclined planes. We may ask
how an object behaves if it rolls down a curved surface, and we may also ask
what happens if we change the value of "g" the free gravitational
acceleration. In fact, in 1696 (shortly after Newton published his "Principia"
in which he announced the Calculus) John Bernoulli asked the following straightforward
question.
If an object were to roll
from a point A to a lower point B along some curve, how long would it take?
What is the curve connecting A to B for which the shortest time is required?
This seemingly simple question
turns out to be fairly deep. In fact, it is closely related to the question
why light "bends" or "refracts" when passing from one medium
(such as air) to another (such as water). Fermat formulated an answer in his
Principal of Least Time, which states that light travels from point A to point
B by a path for which the time of travel is the shortest. Since light travels
at different speeds in different media, the time of travel along a straight
path in one medium is distance/speed and a little reflection (no pun intended)
shows that if it must slow down on entering a new medium, then the straight
line path will not minimize the time of travel.
Bernoulli's problem is similar.
As an object descends along a path under the influence of gravity, its velocity
changes, and the sum of short distances/speeds is not minimized by the straight
line path, but is minimized by a curve called the "brachistochrone"
(which means shortest time). The brachistochrone may be constructed by rolling
a circle along a line without slipping, and tracing the path of a single point
on the circle as it rolls). Later, we'll write down an equation for it. If you
would like to see a demonstration of this construction, press the 
button.
In this part of the Microworld,
you are welcome to experiment with these ideas. We will allow the ball to roll
from the point (0,5} to a point (8,0). So the second point is 5 meters below,
and 8 meters to the right of the first. From what we said earlier about energy
and reciprocity, it is probably not a surprise that the speed at the end of
the trip is always 
meters per second, no matter what the curve. Although it must be kept in mind
that it is not a surprise because we assume that energy is conserved. We have
not and cannot prove that elementary observation about Nature until we develop
the Calculus. So that will have to wait awhile.
Exploration: The Bernoulli slider
In order to create paths,
you should first check what type of curve you want to use to connect the points
(0,5) and (8,0). You may do that by checking one of the four types:
Just click on the type you want. If you click Line, the unique line connecting
the two points will be drawn. If you click Polynomial, then you should also
type in the yellow field: 
how many intermediate points you want the polynomial to pass through. You will
select those points, as we'll explain, later.
You may have between 1 and
7 intermediate points. and the polynomial will pass through those, and the points
(0,5) and (8,0). Be careful here. The ball will stop rolling when it reaches
the 0 height. If your polynomial dips below 0, as it well might, then you should
try again. If it enters a valley that it cannot climb out of (conservation of
energy) then it will oscillate back and forth until you press Esc.
If you select Brachistochrone,
the brachistochrone curve will be drawn passing through the points (0,5) and
(8,0). The general form of this curve, ignoring scaling and translation factors
is, for those interested,
and
If you select Arc of Circle then the unique circle tangent to the x axis and passing through the points (0,5) and (8,0) will be drawn.
Once you select the type,
click on 
to see the curve. If the type was polynomial, the cursor will turn into a cross-hair.
You must then click the appropriate number of times on the graph window to select
the intermediate points. The system will choose the nearest points to the ones
clicked with integer coordinates. The reason is that it will print the exact
rational expression for the function whose graph is being drawn, and this would
be messy if the points were not lattice (integer) points. But be careful to
select points that will give distinct x coordinates. Otherwise you'll get an
"integer divide by 0" message when it attempts to plot.
Finally, press the 
button to do the experiment. The dials will tell you, as usual, how quickly
the ball is moving, and how long it takes. You may also modify the free gravitational
acceleration if you wish.