Uncertainty Principle application to macroscopic particles

[qban88]

1. Homework Statement

The figure shows 1.0*10^-6 m diameter dust particles in a vacuum chamber. The dust particles are released from rest above a 1.0*10^-6 m diameter hole, fall through the hole (there's just barely room for the particles to go through), and land on a detector at distance d below.
If the particles were purely classical, they would all land in the same 1.0 micrometer diameter circle. But quantum effects don't allow this. If d = 1.3 m , by how much does the diameter of the circle in which most dust particles land exceed 1.0 micrometer ?
PART B) Quantum effects would be noticeable if the detection-circle diameter increased to 1.7 micrometers. At what distance would the detector need to be placed to observe this increase in the diameter?

2. Homework Equations
h/2 <= ΔxΔp
Vf = (2*g*d)^(1/2) for a free fall body starting at V = 0.
p = mv
3. The Attempt at a Solution
If I understand it correctly, the suggestion to this problem posted previously in the forum (https://www.physicsforums.com/showthread.php?t=352999) does not work. I tried using λ=h/p finding p by using the velocity of a free fall body given displacement d and λ would be the scattering but it does not work.
I also tried using the uncertainty principle assuming the uncertainty in Δx is 1.0 micrometers initially and from there I can find Δp and therefore Δv = 3.315*10^-13 but I don't know how to relate this to the distance traveled d in order to find Δx final which is what is being asked. Does the uncertainty in velocity increases as the particle moves and if so, in what fashion? Should I just do this as a particle diffraction problem? Having the formula or idea for part A would yield the solution to part B i believe.
Any help would be appreciated

 

[gomboc]

I think the previous suggestion was looking at the problem wrong. They make a point of wanting you to use the uncertainty principle, so here's what I think they want you to do.

Using classical mechanics, you can determine how long each particle is in free-fall before hitting the detector. What you don't know is the particles velocity parallel to the detector surface! You know it's position in that direction to a precision of 1 micrometer, so you can use the uncertainty principle to determine the uncertainty in their velocity in the same direction. (the mass of the particles will cancel out at some point in these calculations if you keep everything in algebraic form up until the final solution).

That velocity should allow you to find some horizontal distance traveled by the particle, which becomes the radius of the circle in which the dust particles land. Once you do this, the second part of the question is just the same problem, but in reverse.

 

[turin]

projectile motion strikes again!