Re: vector quantities and energy

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

Hi --

At 09:22 AM 1/21/99 -0500, GARY HEMMINGER wrote:
> Since we study momentum before energy I've had kids ask me the
> question - why isn't energy (particulary kinetic energy)  a vector?
> It strikes me that this is a great question because it can be
> addressed at so many levels.  Feynman would no doubt say something
> rather different to his colleagues than I would to my weakest 10th
> graders and so on.     How would you answer this?

I'm not sure whether you were asking for the Feynman answer, or for
something that would work on the *weakest* of the 10th graders.

In any case, here's my answer:

***** Executive summary:

Energy and momentum are profoundly related.  They are the two parts of a
larger item.  Energy is just the name that we give to the part of this item
that is unaffected by rotations.

***** Details:

0) We start from the principle: "learning proceeds from the known to the
unknown."  So by way of background, let's consider plain old rotations.
Build a 3D model of a village by gluing small blocks to a piece of
cardboard.  Let there be a school, a postoffice, and a barbershop.  Let the
barber's residence be upstairs, right above the barbershop.

1) For starters disregard the third dimension and consider the model to be
just a fancy map of the village.

Now the main point of vectors is that certain relationships between vectors
are invariant under rotations.  In particular, the length of the vector
from the school to the postoffice is unchanged by a rotation (in the
horizontal plane) of the map.  The north-south component of the vector
changes, and the east-west component of the vector changes, but its length
is invariant.  Similarly the angle between the school-barbershop vector and
the school-postoffice vector is unchanged by a rotation of the map.  (The
story up to this point has followed Feynman's approach rather closely.)

2) We know that there exist three-dimensional vectors.  The interesting
thing about rotations in two dimensions is that the third vector component
is unchanged by rotations in the horizontal plane.  Not only is the length
of the whole three-dimensional vector unchanged, one of its special
components is unchanged by this special rotation.

3) On the other hand, if we consider more general rotations, it is possible
to rotate the model in a non-horizontal plane so that the barber's
residence (which used to be above the shop) is now north of the shop.

4) The analogy to momentum and energy is this:  There is a four-dimensional
vector of which energy is one component, and momentum is the other three
components.  We know that ordinary three-dimensional rotations mix up the
components of momentum with each other, but leave the energy unchanged.
This is quite analogous to the way that rotations in the horizontal plane
leave the height component unchanged.

5) It turns out that there *IS* a generalized "rotation" -- called a
*boost* -- which does mix the energy component in with the momentum
components.

What is a boost?  It is a change in the observer's velocity.  This is
analogous to a rotation, which is a change in the observer's direction of
orientation.

Suppose there is a baseball flying northward. If it flies past you while
you are running in the southbound direction, its energy in your frame is
higher.  On the other hand if it flies past you while you are running in a
northbound direction, its kinetic energy in your frame is lower.  So we see
that the boost (the change in your running velocity) has changed the ball's
energy in a way that depends on the ball's momentum.  The analogy to
rotations is profound.

Summary: Energy and momentum are the two parts of a larger item.  Energy is
just the name that we give to the part of this item that is unaffected by
rotations.


Cheers --- jsd