Calculations based on Neutrons in a Gravitational field

[TanX]

Hello everybody! I am TanX. I was reading about neutrons in a gravitational field, which was based on the Grenoble experiment ( Institute Laue - Langevin ) conducted in 2002. I have put a link down here to the research papers below ( Refer to page number 17 in the booklet for the important diagram )
https://www.physi.uni-heidelberg.de/Publications/dipl_krantz.pdf

Homework Statement

Q: [/B]The neutrons enter the cavity with a wide range of positive and negative vertical velocities, v_z. Once in the cavity, they fly between the mirror below and the absorber above.
The neutron transmission rate N(H) is measured at the detector D ( WHERE H IS THE DISTANCE BETWEEN ABSORBER AND REFLECTOR ) We expect that it increases monotonically with H. Compute the classical rate N(H) which is measured at the detector (D), assuming that neutrons arrive at the cavity with vertical velocity v_z at height z being all values of v_z and z being equally probable. Give the answer in terms of p, the constant number of neutrons per unit time per unit vertical velocity per unit height that enter the cavity with vertical velocity v_zand at height z

Homework Equations

Conservation of Energy.
Properties of Elastic collisions
∫(1-x)^1/2 dx = {2(1-x)^3/2}/ 3
This is probably what I need to use to get to the solution but I have no clue what it means or how it can be used.
3. The Attempt at a Solution [/B]
Okay...Here's my attempt at the solution
So obviously the rate of transmitted neutrons entering at height z is proportional to the range of allowed velocities. Also the given constant (p) can be used.
To find the range of velocities entering at height h
Total energy of neutron at height z = Maximum Potent ( occurs at max height )
(0.5 M v_z²) + MgZ ≤ MgH
Which upon solving gives me
-√2g(H-Z) ≤ v_z ≤ √2g(H-Z)
So now...
d(N(z))/dz = p {v_zmax - v_zmin}
⇒ d(N(z))/dz = 2p√2g(H-z)

After this...I tried integrating the above equation and that's where that given expression is useful... and I don't know how to use this. I would be happy if somebody could assist me here.
Any help will be appreciated! Thanks in Advance!

 

[mfb]

Can you show the integral you set up?

Did you draw a sketch of the z, v_z plane to see how your range of transmitted neutrons looks like?

 

[TanX]

Can you show the integral you set up?

Did you draw a sketch of the z, v_z plane to see how your range of transmitted neutrons looks like?

Here it is :
The experimental setup :

The experiment consists of a setup, consisting of an opening W, the neutron mirror M at H=0 , the neutron detector D, and the Earth's gravitational pull acting downwards. The beam of neutrons flies with constant horizontal velocity v_x from W to D through the cavity between A and M at height h = z. All that reach surface M are reflected elastically. The detector counts the transmission rate N(H).
Here's the v_z and v_x plane diagram:

Sorry for the late reply... I was travelling. Thanks for replying!

 

[mfb]

That is neither the integral nor a sketch of the z, v_z plane.

√2g(H-Z) ≤ v_z ≤ √2g(H-Z)

Here is the relevant equation for the plane.

 

[TanX]

That is neither the integral nor a sketch of the z, v_z plane.Here is the relevant equation for the plane.

Uh...I did not do that... I was given the diagram and asked to classically compute this... So I did not sketch the asked figures... By the way could you tell me how to sketch them? I could make an attempt.
Thanks in Advance!
(Sorry for the late reply..had an urgent exam preparation[emoji28])

 

[mfb]

Make a diagram with two axes, label them, then mark a few points on the boundary between "gets absorbed" and "does not get absorbed" according to the equation you have already.