Suppose that you drop BBs onto a bull’s-eye marked on the floor. According to the uncertainty principle, the BBs do not necessarily fall straight down from the release point to the center of the bull’s-eye but are affected by the initial conditions. 

(a) If the location of the release point is uncertain by an amount Δx perpendicular to the vertical direction and the horizontal component of the speed is uncertain by Δv_x, derive an expression for the minimum spread ΔX of impacts at the bull’s-eye if it is located a distance y₀ below the release point.

(b) Modify your result in (a) to include the effect on ΔX of uncertainties Δy and Δv_y at the release point.