I'm working a problem out of d'Inverno's "Introducing Einstein's Relativity", and I'm hitting a funny issue with my algebra. The problem states:
An atom of rest mass $m_0$ is at rest in a laboratory and absorbs a photon of frequency $\nu$. Find the velocity and mass of the recoiling particle.
The answers are given in the back of the book as
$$ u=\frac{ch\nu}{h\nu + m_0c^2} \qquad m=\left(m_0^2 + \frac{2h\nu m_0}{c^2}\right)^{1/2}. $$
I've found that I can figure out the velocity by starting with conservation of momentum and energy
$$ \frac{h\nu}{c} = \gamma m_0u \qquad h\nu + m_0c^2 = \gamma m_0c^2, $$
(where of course $\gamma=\left[1-\frac{u^2}{c^2}\right]^{-1/2}$) then eliminating the relativistic mass $\gamma m_0$ and solving for $u$. Strangely though, in my first couple of attempts I started with conservation of only energy or momentum, and obtained answers close to, but not quite the same as, the above:
$$ u=\frac{c(2m_0c^2h\nu+h^2\nu^2)^{1/2}}{h\nu + m_0c^2} \qquad (\textrm{from conservation of energy}) $$
$$ u=\frac{ch\nu}{(h^2\nu^2+m_0^2c^4)^{1/2}} \qquad (\textrm{from conservation of momentum}) $$
Is there simply some mistake in my algebra that I haven't managed to suss out, or is the error in assuming that I can proceed from only one conservation law?
As for the mass, I attempted to simply substitute the velocity $u$ into the relativistic mass $\gamma m_0$, but quickly got a monstrosity of $m_0$'s, $h$'s, $\nu$'s, and $c$'s that bore no resemblance to the answer. Is this the correct way to approach this part of the problem, or should I start from some other relation?
EDIT:
Sofia answered the second half of my question regarding the mass of the atom. However, I'm still curious about the first part. That is, why do I obtain different results for the velocity with different equations? One approach may be more difficult or roundabout than another, but if they're all based on the same physical principles, I feel that I should get the same result regardless of which relation I begin with.