🚀 go-pugleaf

> Does anyone have a way of describing the difference between [conceptual
> explainations of energy and momentum] to students?

For me the clearest conceptual relationship between energy and
momentum is seen in the relativistic relationships. In relativity,
energy is to momentum, as time is to space. 

Momentum is associated with changes (in a particle's position) in
space, (and hence must be a vector, since the difference between two
points in space is a vector), while energy is deeply connected to
changes (of a particle or system) in time.

This is a boiled down version of a deep theorem of physics proven by a
brilliant young woman at the turn of this century: Emmy Noether.
Noether's theorem shows that conservation of *momentum* can be
_derived_ by assuming that (the laws of) physics do not change from
one point to another in *space*, while conservation of *energy* comes
from physics being invariant in *time*.

In Newtonian physics,  KE = p^2/2m = (mv)^2/2m
However in relativity the relationship between E and p is:

(1)	m^2 c^4 = E^2 - c^2 p^2

Here E contains both KE and rest mass, so that if the particle is
sitting still (p=0):

	m^2 c^4 = E^2 --> E = mc^2

Eq. (1) has the same form as the "distance" equation in spacetime.
In 3-dim. *space*, distance is given by Pythagorus:

         Distance^2 = x^2 + y^2 + z^2

but in 4-dim. *spacetime*, distance is 

         Distance^2 = c^2 t^2  -  x^2   -   y^2   -    z^2

Compare:  m^2 c^4   =  E^2 - c^2 p_x^2 - c^2 p_y^2 - c^2 p_z^2
eq. (1)

Hope this helps.  --Jim

--
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      James E. Hetrick            Institute for Theoretical Physics
     hetrick@phys.uva.nl              University of Amsterdam     
                                          The Netherlands