The expectation number of collision to slow down below certain value

Question

In Nuclear Physics, a neutron with energy $E_0$ collides with stationary atom of which atom number is A, the neutron scatters isotropically. Then, the very probability density function of afterward energy $E_1$ is $$P(E=E_1)=1/(1-\alpha)E_0$$if $\alpha E_0 \le E_1 \le E_0$ (otherwise the probability is zero), where $$\alpha=(\frac{A-1}{A+1})^2$$ That is, $$E_1/E_0 \sim U[\alpha,1]$$ In system, there are a lot of atoms so the neutron can collides over and over. I want to find how to get the expectation number of colliding to make the energy of neutron being less than certain value, say, $E_t$.
For hydrogen which has atom number of 1, $\alpha=0$ so the case is relatively simple. I found the expectation number in this case by integration is $1+ln(E_0/E_t)$ When A>1, however, the case is too complicated for me to solve this problem. Unlike A=1, in each step there is a positive lower bound of E>0 region so it step by step yields cases to consider.
I googled to find the probability density function provided collided n times and felt very disappointed since I cannot get a way to deal with this function. ref
Does anyone know how to find the expected number?