Re: Scientific method was physical pendulums/ an opportunity

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Leigh,

Regarding,
> A golfer who can hit a ball a maximum distance of 215 m
> on level ground, stands at the edge of a 50 m vertical
> cliff and hits a ball out to sea so that it starts with
> an angle of 40 degrees to the horizontal. At what
> distance from the base of the cliff does the ball land?
> (Neglect air resistance and use g = 9.81 m/s^2.)
>
> What's the thrill in this dull problem? I never before
> noticed that the answer does not depend upon g!

I'm glad you noticed this phenomenon.  I had the same "aha" experience
for a similar problem a number of years ago (although I think my
experience was significantly less intense than your's) for this class
of ballistics problem.  In my case it was for a favorite problem of
mine where one has a small frictionless mass sliding off of the top of a
smooth hemispherical-shaped dome (whose exposed flat side/equator is
attached to the flat ground).  The problem is to predict the angle on the
dome for which the mass leaves contact with the dome and to predict the
landing spot on the ground measured from the edge of the dome in units of
the dome's radius.

It ends up that the class of ballistics problems for which the answer
does not depend on g is that class that has the given geometric
conditions of the problem determine the geometry of the relevant
parabola trajectory without doing any physics.  In the case of your
problem there happens to be a unique (up to arbitrary translations)
inverted parabola which has a chord of symmetry (i.e. horizontal chord)
of length 215 m connecting two points on the parabola whose corresponding
angles of inclination w.r.t. horizontal being 40 deg.  Once the parabola
is found its intersection point with another horizontal cord separated
by 50 m from the first one is determined. Since this is purely a geometry
problem g doesn't come into it.  Similarly with my problem the sliding
mass leaves the dome beginning a parabolic path whose initial height,
slope, and curvature osculate its intersection with the projected circle
of the dome.  Since those 3 numbers uniquely determine the parabolic
trajectory, the landing spot is thus determined once the leaving point on
the dome is determined (which also is independent of g -- but for a
different reason).

David Bowman
dbowman@georgetowncollege.edu