Re: [Phys-L] f'(t) / g'(t) question

[Carl Mungan <mungan@usna.edu>]

Others have already given the answer. Note that g’/f’ = dg/df, which eliminates t as an explicit variable. This is a useful trick for solving differential equations.

This trick can be generalized to partial derivatives. Here is a neat example with obvious relevance to thermo:

http://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/PartialDerivativeVanDerWaals.pdf 

What’s interesting about this example is that the method of solution I present here, which I would argue is relatively straightforward and logical for a physicist, drives mathematician’s crazy. They insist on much more convoluted (call it rigorous if you don’t want to appear partisan) approaches. Go look up the original paper I reference at the beginning of this PDF to see what I mean. -Carl

> On Apr 8, 2016, at 9:52 AM, Lulai, Paul <plulai@stanthony.k12.mn.us> wrote:
>
> Hi.
> I haven't really used my calculus for quite some time.  I was asked by a
> math department colleague the following:
> if  x=f(t) and y = g(t) then dx/dt = f'(t) and dy/dt = g'(t) would be
> velocity vectors.
> so far so good.
>
> The question:
> is g'(t) / f'(t)  a velocity value? can it be used to determine
> instantaneous velocity values?
>
> My knee-jerk response is that it is not a velocity value. However, i don't
> know if there is a way to use this to find the instantaneous velocity. I am
> pretty-much brain-locked into x' ^2 + y'^2 = v^2 when dealing with just
> kinematics.
>
> Is there a useful or common thing that is appropriate here?
>
> Thanks for your time.
>
> Paul Lulai
> Physics Teacher
> St Anthony Village Senior High
> St Anthony Village MN 55418
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Carl E. Mungan, Professor of Physics  410-293-6680 (O) -3729 (F)
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