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[Phys-L] Fraction of energy carried off



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 $B&V (J. What fraction of the total kinetic energy (after the decay) does each particle have? (This is a classical, non-relativistic treatment.)

First, feel free to do the problem before scrolling down to see my answer below. I tried a couple different methods and show only the one that I like the most!

Next, notice what this means about how the fraction of total kinetic energy which each particle carries off depends on the fraction of the total mass which each particle got. Interesting, I think! Seems fundamentally significant to me.

Finally, and the reason I turn to the Forum for Physics Educators, 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
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Solution starts below, last chance to solve the problem first.....
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Let the two masses be m and M, where m/M = $B&V (J. Let their speeds be v and V, respectively.

Conservation of momentum: 0 = p_1(vector) + p_2(vector) ==> p_1 = p_2 $B"a (J p.

Total Kinetic Energy: K = K_1 + K_2 = p_1/(2m) + p_2(2M). We seek the ratios K_1/K and K_2/K :

K_1/K = K_1/(K_1 + K_2)

= (p/(2m)) / ( p/(2m) + p/(2M) )

= 1/(1 + m/M)

= 1/(1 + $B&V (J),

K_2/K = 1 - 1/(1 + $B&V (J)

= (1 + $B&V (J)/(1 + $B&V (J) - 1/(1 + $B&V (J)

= $B&V (J/(1 + $B&V (J).

Multiplying numerators and denominators through by M gives:

K_1/K = 1/(1 + $B&V (J) = M/(M + $B&V (JM) = M/(M + m),

K_2/K = $B&V (J/(1 + $B&V (J) = $B&V (JM/(M + $B&V (JM) = m/(M + m).

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?

An earlier solution used everything in terms of speeds (p=mv, K=(1/2)mv^2) to reach the same results, but took longer to get there. In keeping with that idea, in the relativistic case I'll use the relativistic relationship between energy and momentum,

E^2 = p^2 c^2 + (m c^2)^2,

rather than the formulas relating each to velocity

p = $B&C (J m v, K = ( $B&C (J-1)m c^2,

where $B&C (J = 1/sqrt(1-v^2/c^2).

Here we go:

K_1 = E_1 - m c^2 = sqrt( p^2 c^2 + (m c^2)^2 ) - m c^2

K_2 = E_2 - M c^2 = sqrt( p^2 c^2 + (M c^2)^2 ) - M c^2.

Yes, it's the same momentum magnitude p in both cases, since momentum is conserved. This should result in some simplification, just as in the non-relativistic treatment, even though M and m are different. To proceed,

K_1/K

= [ sqrt( p^2 c^2 + (m c^2)^2 ) - m c^2 ]
------------------------------------------------------------------
[ sqrt(p^2 c^2+(mc^2)^2) - mc^2 + sqrt(p^2 c^2+(Mc^2)^2) - M c^2 ]

= [ sqrt( p^2 + m^2 c^2 ) - m c ]
------------------------------------------------------------
[ sqrt( p^2 + m^2 c^2 ) - m c + sqrt( p^2 + M^2 c^2) - M c ]

= m c [ sqrt((p/(mc))^2 + 1) - 1 ]
---------------------------------------------------------------------
m c [ sqrt((p/(mc))^2 + 1 ) - 1 ] + M c [ sqrt((p/(Mc))^2 + 1 ) - 1 ]

= m [ sqrt((p/(mc))^2 + 1) - 1 ]
-----------------------------------------------------------------
m [ sqrt((p/(mc))^2 + 1 ) - 1 ] + M [ sqrt((p/(Mc))^2 + 1 ) - 1 ]

= m [ sqrt((p/(mc))^2 + 1) - 1 ]
-----------------------------------------------------------------
m [ sqrt((p/(mc))^2 + 1 ) - 1 ] + M [ sqrt((p/(Mc))^2 + 1 ) - 1 ]

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.