Re: Why work before energy in texts

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

At 08:51 PM 10/13/01 -0400, Chuck Britton wrote:

> kinematics
....
> 1/2 Vf^2 - 1/2 Vi^2 = a d                               (eq 1)
>
> multiplying  through by mass gives us
>
> delta KE = F d                                  (eq 2)
>  which is a 'pretty good' intro of the Work Energy Thm.

0) I like this.  Now we are getting somewhere.  I assume
        Vi := initial velocity
        Vf := final velocity
        d  := distance
        delta, KE, F, a := standard meanings

1) A few days ago we saw attempts to connect "F dot distance" with
potential energy;  alas there was no way to make that correct, not without
throwing out the baby with the bathwater.

Here, in contrast, it is the _kinetic_ energy that is connected with F dot
distance.  This is a different story entirely.  This has the advantage of
being true under certain conditions.

When presenting equations such as the foregoing, it is important to clearly
communicate the range of validity.  In this case, the foregoing equations
apply only to point particles, or to objects with no internal degrees of
freedom.

Without such a restriction, it is trivial to find counterexamples, such as
skaters pushing off walls, as was discussed at some length on this list in
August 1999.

It should also be mentioned that this assumes a steady force and uniform
acceleration.  You might be tempted to generalize the formula and interpret
it in terms of average force and/or average acceleration, but this would
not be correct in general, especially in dimensionality D>1.  On the other
hand, if you focus on an infinitesimal interval, you can (under mild
assumptions) take the force to be uniform over that interval -- but then
you have to rewrite equation (2) in terms of differentials, not macroscopic
distance nor macroscopic delta KE.

And of course F represents the _total_ force, including
 -- forces due to gradients of potentials,
 -- forces not due to the gradient of any potential,
 -- et cetera.
This should go without saying, except that in the last couple of days
people have been advocating "laws" based on strange subsets of the force.

2) Let's think about the physics here.  As the saying goes, there are some
things that might be true, and some things that are true for sure.

One thing that is true for sure is
        force * time = momentum
or perhaps more precisely
        force * (delta time) = (delta momentum)                 (eq 3)
This comes with a money-back guarantee.

So we see that force is _intimately_ connected with momentum.  The third
law of motion is intimately connected with conservation of
momentum.  Conservation laws are at the heart of physics.  Whenever you see
a conservation law, pay attention.

Since F is connected to momentum, we shouldn't be surprised to find a
connection to _kinetic_ energy.

If you dot both sides of equation (3) with velocity, and take suitable
averages, you get equation (2) again.

3) Speaking of averages:  This is the answer to a question that Ludwik has
asked several times over the last few days:  How to give an elementary
explanation of the factor of 1/2 that appears in the law
        KE = 1/2 m v^2
Answer:  This 1/2 is the same 1/2 that appears in equation (1).  In the
case of steady uniform acceleration, you can derive equation (1) without
calculus just by considering the _average_ velocity, which is 1/2 (Vf + Vi).

4) Regarding the Subject: line of this thread:  Nothing here provides the
slightest support to the notion that work should be introduced before
energy in texts or classrooms.  Energy is the primary and fundamental
quantity.  Energy is the subject of a conservation law.  Whenever you see a
conservation law, pay attention.

Connecting force to momentum is easy;  connecting force to energy is
tricky, if you want to do it with any semblance of correctness.

5) To repeat:  There are important limitations on the validity of equation
(2).  It was derived under the assumption of uniform acceleration of an
object with no internal degrees of freedom.  This is just one of the many
pitfalls of the "work first" approach.  If you elevate equation (2) to the
status of "theorem" and blindly apply it to a skater, or to a thermodynamic
system, it's just wrong.