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From: gwyn@brl-vgr.ARPA (Doug Gwyn )
Newsgroups: net.math
Subject: Re: Calculus problem
Message-ID: <3077@brl-vgr.ARPA>
Date: Sat, 31-Mar-84 02:17:35 EST
Article-I.D.: brl-vgr.3077
Posted: Sat Mar 31 02:17:35 1984
Date-Received: Sat, 31-Mar-84 10:08:51 EST
References: <115@iham1.UUCP>
Organization: Ballistics Research Lab
Lines: 31

Your problem boils down to two assumed initial conditions:

	s(t=0) = 0

	v(t=0) = 0	where v = ds / dt

and one force equation:

	(F = )  m a  =  m g - k v^2	where a = dv / dt = d^2s / dt^2

(sorry about the superscripts)

In terms of s(t) this is a second-order non-linear ordinary differential
equation with two equations of constraint (boundary conditions):

	s(0) = 0

	s'(0) = 0	where ' denotes time-derivative

	m s''(t) = m g - k (s'(t))^2

What you do at this point depends on why you are asking the question.
I myself would just look up the solution and save time.  Since you
have the solution for s(t), all you technically need to do is just
demonstrate (by substitution) that it satisfies the three constraints.
If this is for a course in ODE, you should use some of the tricks
taught in the course to methodically synthesize the answer.

Note that IF there is a limiting velocity, its value is trivial to
derive, since then a == 0 and the force equation is trivial to solve
for v.