Tina wrote:
Still waiting for the conservation of momentum derivation. My husband,
also trained as an engineer, casually remarked he didn't think you
could get from Newton's First Law to the that confirms my memory,
but we are both willing to have that belief rebutted.

You can't get conservation of momentum from F = m*a (or vice versa) since
the latter is not a statement about anything being conserved. But you can
get "For every force there is an equal opposite force" from conservation
of momentum, and vice versa, with a small number of assumptions. You can
use derivatives to derive one way and integration to derive the other.
Here are several conservation laws that share a common derivation,
starting with:

Center of Mass is Conserved
---------------------------
Center of mass of a closed system of particles of mass m1, m2, m3, ... mn
must remain fixed for all time, which with a suitable selection of
coordinate origins may be stated mathematically as:

(a) m1*x1 + m2*x2 + m3*x3 + ... = 0
m1*y1 + m2*y2 + m3*y3 + ... = 0
m1*z1 + m2*z2 + m3*z3 + ... = 0

Note that this doesn't say that, for example, x1 can't vary with time. It
only says that if it does then m1, m2, m3, x2 or x3 or other masses or
positions must somehow change so the left hand side still remains zero.

Momentum is Conserved
---------------------
If position with respect to time is continuous (no discontinuities; e.g.
no jumps) then we can take the time derivative of the above, yielding:

(b) m1*dx1/dt + m2*dx2/dt + m3*dx3/dt + ... = 0
(And so on for the other coordinate axis.)

This is of course just the conservation of momentum equations because
dx1/dt = Vx1, a velocity. Note that d(m1*x1)/dt would have been more
appropriate if the mass of particles varies with time.

Force is Conserved
------------------
Given the continuity assumption above, then we can keep taking time
derivatives of the above, yielding the next conservation statement:

(c) m1*d^2(x1)/dt^2 + m2*d^2(x2)/dt^2 + m3*d^2(x3)/dt^2 + ... = 0
(And so on for the other coordinate axis.)

This is of course just the old equal and opposite action statement in
mathematical form because d^2(x1)/dt^2 = ax1, an acceleration.

So if one claims any _one_ of the above conservation laws exists then the
other two appear to follow with only a small set of (presumably
reasonable) assumptions.