# Nonrelativistic Lab-frame Velocity as a function of kinetic energy per unit mass

Hi everyone--I'm a bit stuck trying to follow a calculation in an article (Nucl. Phys. B360 (1991) p. 145-179) regarding the lab-frame relative velocity of two colliding particles as a function of the kinetic energy per unit mass. (I'll include references to equations in the article, but the article is not required for this particular question.)

Suppose we have two particles of mass $$m$$ colliding with one another with energies $$E_1, E_2$$ and 3-momenta $$\mathbf{p_1}, \mathbf{p_2}$$. Define $$p_1 = |\mathbf{p_1}|, p_2 = |\mathbf{p_2}|$$

Define (eq. 3.3)
$$s = 2m^2 + 2E_1E_2 - 2p_1p_2\cos\theta$$

Which, I believe is the same as the center of mass energy $$(p_1^\mu+p_2^\mu)^2$$.

Now define the kinetic energy per unit mass in the lab frame, (3.20)
$$\epsilon = \frac{(E_{1,\mathrm{lab}}-m)+(E_{2,\mathrm{lab}}-m)}{2m}$$

First question: Why can we write
$$\epsilon=\frac{s-4m^2}{4m^2}$$

Second question: Why is is true that the lab velocity is given by
$$v_\mathrm{lab} = \frac{2\epsilon^{1/2}(1+\epsilon)^{1/2}}{1+2\epsilon}$$

Thanks very much for any assistance!
Best,
Flip

Meir Achuz