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Swimming Hole Problem
A person is jumping into a swimming hole by holding onto a rope
suspended, as from a tree, over the water, swinging out, and releasing
the rope.
After releasing the rope the person has a certain velocity vector V,
travels in a ballistic (parabolic) arc until splashing down into
the water.
The rope is suspended from a point which is h units above
the water, and the rope is of length l.
At the dock, the rope makes an angle THETA from the vertical
(THETA << PI/4); the person releases the rope when it
makes an angle PHI from vertical (PHI < THETA).
Question
At what angle PHI should the person release the rope?
- to maximize distance from dock to landing point?
- to maximize time in free fall?
- to maximize time from leaving the dock to splashdown?
- to maximize apogee height?
In general, describe the path in closed form.
Note that the first part of the path is circular, the second
part is parabolic.
Please don't say PHI should be 45° since THETA
isn't even that much.
Make any simplifying assumptions. Assume the person is spherical
and point-size, the rope is rigid and massless, sine(THETA) = THETA, and so forth. Ignore
air resistance. Assume a constant gravitational acceleration, and a non-rotating earth.
To start, assume the height h is equal to the length l,
that is the center of mass of the person skims the water at the
bottom of the pendulum swing.
Please let me know.
I don't have any solution to this problem.
This problem was posted to alt.math.recreational
on January 25, 1999, as message
<[email protected]>.
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