relativistic energy and momentum questions.

MathematicalPhysicist

problems statement:
1. a nucleus of mass m initially at rest absorbs a gamma ray (photon) and is excited to a higher energy state such that its mass now is 1.01m, find the energy of the incoming photon needed to carry out this excitation.

2. A moving radioactive nucleus of known mass M emits a gamma ray in the forward direction and drops to its stable nonradiactive state of known mass m.
Find the energy E_A of the incoming nucleus such that the resulting mass m nucleus is at rest. The unknown energy E_c of the outgoing gamma ray should not appear in the answer.
attempt at solution
1.well, for the first question i think this is fairly simple:
from conservation of 4-momentum we have before 4-momentum is:(mc,0) after
(E/c+E_ph/c,P) so we have : (mc)^2=(E/c+E_ph/c)^2-P^2=(E/c)^2-P^2+2EE_ph/c^2+(E_ph/c)^2=(1.01mc)^2+2EE_ph/c^2+(E_ph/c)^2 where (E/c)^2-P^2=(1.01mc)^2, here im kind of stuck with E which is not given, any hints?

2.for the second the answer in the book is E_A=((M^2+m^2)/2m)c^2
but i dont get it, here's my attempt to solve it:
the before 4 momentum is (E_A/c,P) after: (E_c/c,0)+(mc,0)=(E_c/c+mc,0)
which by the square of the momentums we get that:
(E_A/c)^2-P^2=(E_c/c+mc)^2=(Mc)^2 but im not given P so im kind of stuck here again, i thought perhaps calculate it in the rest frame of M which means that the before is:
(E_A/c,0) the after is (E_c/c,0)+(E/c,-P) but still don't get far with it, any help is appreciated, thanks in advance.

Hootenanny

For question one, what is the 'rest energy' of the nucleus?

MathematicalPhysicist

well, if it wans't clear in my post, obviously it's mc^2, and i wrote in 4 momentum notation (mc,0) for the before the absorption of the photon.

Hootenanny

Perhaps I'm missing something here, but couldn't you write;

[tex]p^2 = (mc)^2 - (1.01mc)^2[/tex]

MathematicalPhysicist

well first, it should be minus that ofocurse cause this way we get a negatrive value where everything there is positive.

and im not sure, what's wrong with what i wrote, first we have (mc,0) after that we have the absorption: (E/c+E_ph/c,p) now (E/c)^2-p^2=(1.01mc)^2 and
E_ph=mc-E/c=mc-sqrt((p)^2+(1.01mc)^2) but how do you find p?

MathematicalPhysicist

i think that p=E_ph/c, am i wrong?

MathematicalPhysicist

ok, i solved question number 2.

MathematicalPhysicist

any news on question number 1?