Suppose you drop little balls of mass m on to a target marked on the floor. According to the uncertainty principle, the balls should not necessarily fall straight down even if you attempt to hold them precisely above the target and release them with no horizontal motion. Rather, the initial conditions can be modified by quantum considerations. Let 8x the accuracy to which you determine the horizontal of the position of the ball. a) Let Yo be the height at which the balls are released. In terms of 8x, Yo, m and g, determine the minimum horizontal spread AX of the balls at the target. b) Then suppose you decide to make a strategy to trade off between uncertainty in position and momentum to minimize the sum of these two con- tributions to aim as precisely as possible. What is the expected deviation from the target? For this question, you may neglect the effects of the uncertainty principle in the vertical direction, but you might enjoy thinking about how to modify your answer should you include those effects.