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Re: Could you survive the ride?



Brian's admonitions wrt the limited applicability of models are hereby
duly noted and seconded.

Nevertheless ...

In your x = t^(5/3) case, the force itself has a clearly unreasonable
behavior at t=0 and yet the particle moves in an apparently reasonable
fashion.

If you are interested in *really* pathological mathematical behaviors, may
I suggest that you consider the Newtonian solution for the motions of
particles subject to the also unphysical but at least *seemingly*
well-behaved forces

F1 = +(1 N/m^2) x^2

and

F2 = +(1 N s^3/m^3) v^3.

In the first case, a 1 kg mass released from rest at x = +1 m has a
mathematically well-defined position for less than 3 seconds. During that
time it reaches arbitrarily large speeds and positions.

In the second case, the same 1 kg mass with an initial velocity of +1 m/s
has a mathematically well-defined position for only half a second. During
that time it reaches arbitrarily large velocities, but only travels a
distance of one meter!

If you want to see the details, they may be found in "The Pathological
Kinematics of Unphysical Force Laws," Am. J. Phys., 60, 238-241, (1992).

John

On Fri, 23 Oct 1998, Paul Goodman wrote:

...
Consider the functions f(t) = t^(7/3) and g(t) = t^(5/3). Let each function
represent the position at time t of a cart moving along a straight track.
Both these functions vanish at the origin, have first derivatives which
vanish at the origin, but have second derivatives which behave much
differently at the origin -- f" vanishes, while g" goes belly up.
...

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