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[Phys-L] Re: (now fixed) RE: Fraction of energy carried off



At 04:14 PM 10/9/2005, Ken Caviness wrote:

I have a physics question for you:

(modified from Young & Freedman, 11th ed., 8.88): A large nucleus at rest
decays into 2 particles the ratio of whose masses is r. What fraction of
the total kinetic energy (after the decay) does each particle have? (This
is a classical, non-relativistic treatment.)
/// consider
doing the same problem relativistically. I don't get the same simple, neat
answer, in fact, I get a complicated mess. Does this still simplify
somehow? Or is the peculiarly simple and appealing result only true in the
classical limit? Maybe a different (and still elegant) result is always
true? I've tried total energies instead of kinetic energies, still a mess.
Besides, that wouldn't work in the classical limit anyway. Sigh! I would
appreciate any thoughts on this.

Thanks,

Ken
///
A nifty result. The fraction of kinetic energy each got is the same as the
fraction of the mass that the _other_ got. For instance, if an object
explodes into 2 pieces, getting 1/10 and 9/10 of the mass, respectively,
then they carry away 9/10 and 1/10 of the energy of the explosion,
respectively. A useful corollary lets you treat any collision/rebound of
two objects: shift into the center of mass coordinate system, in which the
incoming relative momenta are equal in magnitude, and in that coordinate
system the above fraction of energy relationships will hold. Nice, don't
you agree?

///

Nothing like M/(M+m) showing up here, and I don't see how to proceed without
using classical limit approximations, for example. Maybe the result is just
a lucky coincidence? I would be disappointed if that were all there was to
it.


A pretty classical result for any two objects departing a stationary
joint object - you could mention that their directions are on a single
line of flight. But if you want a quantum approach, ducking the
classical "mass deficit" avoids the issue somewhat, doesn't it?



Brian Whatcott Altus OK Eureka!