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Re: Physics Answer to a Question...



On Sun, 9 Feb 1997, A. R. Marlow wrote:

In spite of several claims to the contrary, this problem does NOT reduce
to the standard rocket problem, even when the old man's mass is allowed
to diminish. The reason is that the standard derivation of the rocket
equation requires taking a limit delta m --> 0, where "delta m"
represents the small amounts of mass being ejected. In the present
case the limit cannot be taken, and, as pointed out in an earlier post,
the resulting nonzero additional term is approximately of the same
magnitude as the "rocket" term. When both are in effect (i.e., when
the mass is allowed to diminish), only about 5000 "spits" are needed
to get to the desired 1 m/s.

I disagree. No need for any limits (although in this case it doesn't
matter.)

Let's apply conservation of momentum to the nth spit in the frame of the
earth:

Momentum before = Mv

where M = mass of the old man before the nth spit [(50 - (n-1)*0.001) kg]
and v = speed of the old man before the nth spit

Momentum after = (M - m)(v + dv) - m(V - v - dv)

where m is the spit mass [0.001 kg], dv is the old man's added speed (not
infinitesimal), and V is the speed of the spit relative to the old man
(5 m/s)

Setting initial and final momenta equal we find

dv = mV/M

just like for the rocket except that dv and m are not infinitesimal.

Now this is easy to set up on a spreadsheet with M reduced by one gram for
each spit and the resulting dv added to the current v after each spit. I
have done so and the result is 9061 spits to reach 1 m/s. This is exactly
the same answer that is obtained using the rocket formula. Note: There
*is* a difference between the discrete case and the rocket case, but that
difference is less than 1 gram, hence the same number of spits.

John
----------------------------------------------------------------
A. John Mallinckrodt email: mallinckrodt@csupomona.edu
Professor of Physics voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223
web: http://www.sci.csupomona.edu/~mallinckrodt/