Question

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 ax perpendicular to the vertical direction and the horizontal component of the speed is uncertain by av derive an expression for the minimum spread axof impacts at the bull's-eye if it is located a distance yo below the release point. (b) modify your result in (a) to include the effect on axof uncertainties ay and av, at the release point.

Answer 1

The uncertainty principle states that position and momentum cannot be simultaneously measured with perfect accuracy. To derive the minimum spread of impacts at the bull's-eye, uncertainties in the initial conditions and the horizontal velocity component need to be considered. The effect of uncertainties in vertical distance and the horizontal velocity component can be included in the calculation.

Step-by-step explanation:

The uncertainty principle states that the position and momentum of a particle cannot be simultaneously measured with perfect accuracy. In this scenario, the bbs do not fall straight down from the release point to the center of the bull's-eye due to the uncertainties in the initial conditions.

(a) To derive an expression for the minimum spread Ax of impacts at the bull's-eye, we can consider that the bbs are released with an uncertain velocity av in the horizontal direction. The time of flight for the bbs can be calculated using the vertical distance yo and the vertical component of the velocity av. From this, we can calculate the uncertainty in position Ax.

(b) To modify the result in (a) and include the effect of uncertainties ay and av at the release point, we can consider the uncertainties in the vertical distance ay and the horizontal component of the velocity av. This would introduce additional uncertainties in the time of flight and the position of the impacts.

Answer 2

The expression for the minimum spread ax of impacts at the bull's-eye, taking into consideration the uncertainties in the location of the release point and the horizontal component of speed, is ax = h/4m√(2gyo).

Step-by-step explanation:

In order to derive an expression for the minimum spread ax of impacts at the bull's-eye, we need to consider the uncertainty in the location of the release point (ax) and the uncertainty in the horizontal component of the speed (av). Assuming that the bull's-eye is located a distance yo below the release point, the minimum spread ax can be calculated using the uncertainty principle.

Using the uncertainty principle, we know that the uncertainty in position (ax) and the uncertainty in momentum (av) are related by the equation AxAp ≥ h/4. Rearranging this equation, we can solve for the uncertainty in position Ax = h/4Av. Substituting Av = m√(2gyo), where m is the mass of the bb, g is the acceleration due to gravity, and yo is the vertical distance between the release point and the bull's-eye, we get Ax = h/4m√(2gyo). Therefore, the expression for the minimum spread ax is ax = h/4m√(2gyo).