[Phys-L] Re: Momentum

[John Denker <jsd@AV8N.COM>]

David Abineri wrote:
> In the ideal case of two equal masses colliding linearly where one is

Nitpick:  Some people would prefer to say two _objects_ colliding.
Each object has mass, not the other way around.  Mass is just one
property of the object, not a synonym for object.  Things that aren't
objects can also have mass.

> stationary and one is moving whereby they stick together after the
> collision (totally inelastic), the conservation of momentum leads to the
> conclusion that half the kinetic energy is lost to other forms.  Of
> course, in the real world, one has sound, deflection of materials,
> friction at least that will convert the energy to other forms.
>
> But what does this really say?  What do the "ideal" conditions mean?
> Why, regardless of the masses is exactly half of the kinetic energy
> lost.  When we say "ideal" conditions to we mean frictionless, rigid
> objects in an airless world?

That's an interesting question.

I think giving a general answer to the most general form of the question
is too hard ... but I can explain how to handle interesting specific
cases.

Let's consider the case of two fully-laden freight cars.  One is initially
at rest in the lab frame, while the other is a few dozen yards away and
approaching at 7 mph.

Let's figure out some additional numbers.  The Reynolds number for a freight
car at this speed is large enough that fluid inertial effects will be larger
than fluid viscosity effects.  The air pressure on the front of the car will
be the dynamic pressure (.5 rho v^2) times the coefficient of form drag (which
we can approximate by unity).  Now 7 mph is about mach 0.01, so you can do
the numbers in your head:  all the aerodynamic forces scale like v^2, so the
pressure drag is about 0.0001 atmosphere, or about 0.0015 psi.  There may be
some pressure-recovery on the backside of the car, but probably not a whole lot.

The frontal area is 10'8" by 15'1" or 161 square feet.  So the force due to
form drag will be about 0.24 lb(f) at the given initial velocity.  Given
that the mass is something like 140,000 pounds, the thing will be decelerating
at the rate of 0.0000017 Gees.  That's not zero ... but you're unlikely to
notice it unless you look very closely.

So you begin to see the style here:  I'm not going to make assumptions,
I'm going to make approximations ... preferably _controlled approximations_.

Air drag is not zero, but it can be neglected to a good approximation.  You
can include it as a perturbation if you want an even better approximation.

   BTW, in contrast, for two parallel plates in a viscous fluid, it's not clear
   they ever really "collide" in the classical sense ... because it takes a
   reeeeally long time to squeeze the fluid out from between the plates.

   Also, if the Reynolds number is really low (such as bacteria swimming through
   honey) then it hardly makes sense to apply Newton's laws at all;  it is
   better to apply the Aristotelian laws of motion:  objects at rest remain at
   rest, and objects in motion come to rest.  There is no inertia (to an
   excellent approximation), just friction.

The rail cars are an example of a controlled approximation, controlled by parameters
such as
  -- Reynolds number (which is large)
  -- density ratio, i.e. the average density of the car divided by the density of
   the air (which is large)
  -- fineness ratio, i.e. sqrt(frontal area) divided by car length (which is
   less than 1, certainly not large) meaning the object is not big, flat, and
   thin (like a parachute) ... but neither is the fineness so large that surface
   friction drag becomes dominant.

===========

We can safely neglect friction in the wheels.  I don't know how to calculate
it, but I have observed first-hand that they put really good bearings on
rail cars.  This stands to reason; otherwise the railroad's fuel costs would
go up due to the friction in the wheels ... and perhaps worse, there would
be overheating problems in the wheel bearings.

Specifically, I have observed that if you get a rail car rolling, it will
roll a very long ways with not much diminution of its speed.  (You can easily
observe the low rolling friction of an automobile; rail cars are the same
only more so.)

===========

The half-energy result does not depend on the objects being "rigid" or
"frictionless".  Indeed the objects *cannot* be truly rigid or truly
frictionless.

Having two truly ideal rigid freight cars collide _and stick_ would be
pathological.  (This has been discussed on this list before.)  There needs
to be some non-rigidity, some complexity somewhere to implement the required
dissipation ... otherwise they just rattle back and forth forever.

Real rail couplings have _draft gears_ that explicitly provide the necessary
dissipation.

===========

One requirement that wasn't mentioned is simply that the objects be identifiable
as objects -- without bits and pieces spalling off during the collision.

Also for freight cars, we can safely assume the speeds are low enough that
special-relativity corrections are negligible.

Perhaps the biggest thing to worry about is the possibility that the rails
are not quite level.  But again, this can be handled by a controlled
approximation.  You can measure the inclination of the rails.  If it's
small, you can ignore it, and if it's large, you can account for it.

You begin to see the style:  make a list of every contribution you might
need to worry about.  Then estimate each one.  Often an upper bound on
the magnitude of the contribution suffices to show that it is negligible,
in which case it doesn't even need to be a tight upper bound.  If it's
not negligible, go back and do a more-precise calculation, and include
it as a correction term.
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