Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-L] quickest route



This problem is very similar to the following "skateboard" problem which
appeared in Science News, 12/3/11 on p 10 ("Skateboarders Rock at Physics").
Skateboarders are given two choices. Skate down a simple inclined plane
(Case a) or roll down a "complex" inclined plane (Case b) which angles down
more steeply than Case a in two sections connected by a horizontal section.
Both cases start at the same high point and end at the same low point. The
article maintains that skateboarders who choose Case b will get to the
bottom faster because of their greater speed at the end of the first steeper
inclined plane in that case. This wasn't obvious to me under all
geometrical conditions. I guess this is the "ill-posed" nature of the
problem that has been discussed here. I decided to work out the solution.
I assumed a point particle (so no rotations need be considered) and no
friction. Therefore, as has been pointed out, no calculus is needed; just
the uniform accelerated motion equations on the various inclined planes. I
defined the angle phi as the angle the inclined plane in Case a makes with
the horizontal. Sure enough, Case b has a smaller time to the bottom for
angles phi up to about 36 deg with the biggest advantage as phi approaches
zero. But Case a has a shorter time for phi greater than 36 deg (which is
not mentioned in the Science News article). At phi = 90 deg both times are
again equal. This happens for the same reasons discussed here by John
Decker although this is a somewhat more complicated case because Case b has
two ramps. I actually wrote a letter to Science News describing the need
for considering all the geometrical possibilities. This letter was
published by them in the January 14, 2012 issue in their "Feedback" section.

Don

Dr. Donald G. Polvani
Adjunct Faculty, Physics
Anne Arundel Community College
Retired