This plot of *kinetic energy*
versus *momentum* has a place for most moving objects
that folks encounter in everyday life. It shows objects
with the same kinetic energy (horizontally related) that carry different
amounts of momentum, as well as how the speed of a
low-mass object compares (by vertical extrapolation) to the speed
after perfectly inelastic collision with a larger object at rest.
Highly sloped lines (rise/run = 2) mark contours of constant *mass*,
while lines of unit slope mark contours of constant *speed*.
Parallel contours are separated by intervals of three orders of magnitude,
so objects from small (radio frequency photons) to large (the observable
universe) are included.
The top and right axes are labeled in practical units, while the bottom and
left axes show SI unit values of the quantities being graphed.
The plot further illustrates
where lightspeed, Planck's constant, and *k*T figure in.

Use this graph to quickly estimate the kinetic energy needed to accelerate an object to a given velocity (like a person to earth escape velocity), as well as the kinetic energy transferred in a momentum exchange between two objects of differing mass (like when you hit the planet after falling off the roof). It's also easy to see, for example, that laser weapons with many times the kinetic energy of a speeding automobile have much less recoil e.g. than that of a rifle.

The centered diagonal-line running from
lower left to upper right is a
"curtain of finite lightspeed *c*" that Newton had no
reason to expect. When the curtain is in place
redirecting lines of constant mass asymptotically
onto v=*c*, only the lower right half
of the plot is accessible. For most of that accessible area Newton's
approximations work pretty darn well. Although *coordinate
velocity* v=dx/dt is a dead-end variable near its
upper limit, *proper velocity*
w=dx/dτ and *momentum* p=mw as well as *relativistic energy*
γm*c*^{2} remain useful on both sides of
w=*c*. The resulting slope-change in energy versus momentum is
in practice verified every time someone examines electron-wavelength
as a function of energy, e.g. using the transmission
electron microscope at your neighborhood hospital. Note that
the location of that
electron
elbow is determined by the
metric
conversion factor *c* between
meters and seconds, which also limits the rate at which
photons can move.

Planck's constant *h* determines the
*deBroglie wavelength* (useful for objects of low momentum), and
*k*T yields the **thermal energy:** *per
nat
of state
uncertainty*
as well as
*per pair of quadratic modes* for distributing heat.
Thermal kinetic energy of translational motion in 3D is
therefore about (3/2)*k*T per particle. Note
that the relationship between the energy and momentum (also frequency
and wavenumber) of an object
is called its *dispersion relation*. Such
relations can be plotted for non-vacuum media on this
graph, thus illustrating for example the concepts
of medium-conferred *effective mass* and frequency-dependent
*refractive index*. For
ring
rotators
of given density one can also superpose in the lower
left corner of this plot the freeze-out zone due to angular
momentum quantization, which requires that the rotator's deBroglie
wavelength span the ring's circumference an integral
number of times.

What are some other things that might be fun to plot on such a graph? A partial list begins here...

- A moving snail.
- Myosin molecule moving along an actin muscle fibril.
- 300 keV microscope electron.
- 1/refractive index dispersion profile for light in glass.
- A line for the mass of the observable universe.
- Recoil energy of the earth when you fall off the roof (down near
*k*T).

What else?

One place to start might be Edwin Taylor's notes on
the
principle of least action, which also links to some papers by
Tom Moore. These provide a foundation
*at all education levels* for descriptions of motion
based on the conservation laws for energy and momentum. On simplifying
concepts the late
William
Shurcliff (1909-2006), who self-published a monograph
on proper-velocity (w=dx/dτ here), said
to me some years ago:

```
I wish you had pointed out this neat fact: because proper speed
can be infinite, improper speed is necessarily limited -- cannot
exceed 3x10
```

^{8} m/sec. If someone asks: "Why is the speed of light
limited?", we answer: "It is not limited if defined in a manner
that requires no synchrony, i.e. if defined as a ratio of distance
(defined most simply) and time duration (defined most simply)."

Here find a draft list of equations
for interconverting between an object's kinetic
energy K, momentum p, rest mass m, coordinate speed
(also group velocity) v=dx/dt, and deBroglie
wavelength λ. In effect, these implement the free particle
dispersion relation. Except for wavelength, the basic idea is
to let you calculate any variable from any *pair* of
other variables (like m,v) assuming that γ=dt/dτ
and w=dx/dτ are
defined as usual in terms of v above.

Note also that when v is much less than *c*, *proper
velocity* w is essentially equal to *coordinate velocity* v, and the *kinetic
energy per unit mass* (γ-1)*c*^{2} becomes ½v^{2}.
The above relationships then simplify to their more familiar Newtonian forms:

Useful velocity-parameter conversions that work at any speed (at least in the uni-directional motion case) include:

These various velocity parameters are defined above
in terms of the map-time t, map-position x, and proper-time τ variables
used in the flat-space metric equation (*c*dτ)^{2} =
(*c*dt)^{2} - (dx)^{2}. That velocity angle (or rapidity)
η comes in handy, for example, when considering the motion of accelerated
objects. Of course, an integrative approach to
anyspeed acceleration
first starts with the v less than *c* case:

This pattern of systematic solving can then be adapted for the anyspeed motion variables discussed above, if we replace the familiar coordinate acceleration a above with the proper acceleration α experienced by the accelerated object. Let us know if you find errors in these equations, as this is a first pass at putting them all in one place.

Although we refer to the above as anyspeed equations, some of them may confuse your calculator at low speed because of roundoff errors in the high precision difference between two numbers. This is yet another reason that the simpler Newtonian equations come in handy for most of the accessible area in that kinetic energy versus momentum plot.

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