Re: kinematics, traditional or not

["John S. Denker" <jsd@MONMOUTH.COM>]

Tim Folkerts wrote:
> I don't view kinematics as derived from dynamics, or as a special case of
> F=ma.
>
> The kinematic equations are merely mathematical definitions.  After
> DEFINING
> v==dr/dt and a==dv/dt , you can use calculus to DERIVE  x = 1/2at^2 +
> v(0)t+ x(0), etc for constant acceleration, or DERIVE a = v^2/r for
> circular motion.  This has nothing to do with forces.

An excellent point.  We need restrict this to the
nonrelativistic limit to keep nitpickers at bay, but
otherwise that's real hard to argue with.

> F = ma is a much deeper statement.  You can determine mass; you can
> determine acceleration; you can determine force.

Yes.

> It turns out that, EXPERIMENTALLY, F=ma.

Yes.

> There is nothing that a priori requires this
> particular relationship, it just happens that there is such a nice, simple
> relationship in the universe.  It is conceivable that an experiment might
> disagree with F=ma and that it would have to be modified.  But I don't see
> how you could change a==dv/dt==d^2r/dt^2.

Be careful.  There's a lot of truth in that, but also
a huge loophole.

Indeed in relativistic conditions, F=ma no longer holds.
You can patch things up by redefining F ... or by redefining
a.  And in fact people really do redefine v and a!  We redefine
        v = (d/d tau) x                 (1)
        a = (d/d tau) v
instead of
        v ?= (d/d t) x                  (3)
        a ?= (d/d t) v

and the relativistic momentum is p=mv using the v from
equation (1) not equation (3).

=======================

There's another issue lurking here.  This is a pretty deep
epsistemological issue.

By way of groundwork, let me lay out an analogy.  (Remember
the saying:  learning proceeds from the known to the unknown.)

Given good operational definitions of mass, length, and time,
we can construct a unit of energy.  In the SI system we start
with kg, m, and s and construct the Joule.  Meanwhile, the
world is full of thermometers that measure energy in degrees.
That seems goofy, but that's the way it is.  To patch things
up, we introduce a universal constant, Boltzmann's constant.

This is analogous to the points Tim and Joe have been
making.  Given mass, length, and time, we can construct
a unit of force.  As Tim quite rightly points out, it is
conceptually possible to measure force in a way that is
independent of the conventional definitions of mass, length,
and time.  We most certainly could do that, using a legacy
standard "fish scale" or whatever.  On the other hand, as
I believe Joe had in mind, we could DEFINE force according to
        F = (d/d tau) m v
which DEFINES dynamics to be a consequence of kinematics.

We can reconcile the two viewpoints by introducing a new
universal constant, the Folkerts/Heafner constant.  This
converts from whatever operational unit Tim was using to
measure force, converting it to conventional units.

So I suggest you guys not start a little-endian versus
big-endian feud.  There is no basis in physics for
preferring one approach to the other.  There might be
a pedagogical argument for starting one place or the
other, but probably not a strong argument -- and in any
case, at the end of the year the students ought to be
able to see it both ways.