Beam width uncertainty, deflection

[unscientific]

Homework Statement

In a scanning electron microscope, a 0.3 keV electron beam, collimated by an
aperture of width ∆xi, is directed towards a target 1 m away from the aperture. Assuming a field-free region from aperture to target, the uncertainty ∆xf in the transverse
coordinate of the electron when it hits the target comes in part from the uncertainty
in the initial coordinate and in part from the uncertainty in the transverse velocity
∆vx. What is the typical distance by which the electrons miss the target, given that
the beam is aiming with the highest possible precision? Find the value of ∆xi that
gives the smallest total uncertainty ∆xf. Discuss quantitatively whether your result is
in accordance with electron diffraction of a slit of width ∆xi
. [10]
You now enter a contest in which the contestants drop a marble with a mass of
20 g from a tower onto a small target 25 m below. As for the electrons, find the value
of the uncertainty ∆xi in the initial transverse coordinate that gives the smallest total
uncertainty ∆xf at the ground. Comment on the relevance of your result. [5]

I'm not sure if I'm interpreting the question correctly..

Homework Equations

The Attempt at a Solution

 

[Nate810]

For part 1, the total uncertainty (Δxf) comes from the following: Δxf = Δxi + (Δvx/vx) * xi where vx is the velocity of the electron beam. The minimum total uncertainty (Δxf) will occur when Δxi is at its smallest value, which is 0. Therefore, the typical distance by which the electrons miss the target is equal to the uncertainty in the transverse velocity (Δvx). This result is in accordance with electron diffraction of a slit of width Δxi, since the smaller the width of the slit, the smaller the uncertainty in the position of the electrons. For part 2, we can use the same equation. The total uncertainty (Δxf) comes from the following: Δxf = Δxi + (Δvx/vx) * xi where vx is the velocity of the marble. The minimum total uncertainty (Δxf) will occur when Δxi is at its smallest value, which is 0. Therefore, the typical distance by which the marble misses the target is equal to the uncertainty in the transverse velocity (Δvx). This result is not as relevant in this case, since the marble is not a particle, and diffraction does not apply.