A 0.662-kg basketball is dropped out of a window that is 6.32 m above the ground. The ball is caught by a person whose hands are 1.82 m above the ground. (a) How much work is done on the ball by its weight? What is the gravitational potential energy of the basketball, relative to the ground, when it is (b) released and (c) caught? (d) What is the change (PEf - PE0) in the ball's gravitational potential energy?

F = m*g = 0.662 * 9.8 = 6.49 N.

a. Work=F*d=6.49 * (6.32-1.82)=29.2 J.

b. PE = mg*h = 6.49 * 6.32 = 41.0 J.

c. PE = 6.49 * 1.82 = 11.81 J.

d. 11.81 - 41 = -29.2 J.

To answer these questions, we need to understand the concepts of work and gravitational potential energy.

Work is defined as the force applied to an object multiplied by the displacement of the object in the direction of the force. The work done on an object by its weight is given by the formula: Work = force x displacement x cos(theta), where theta is the angle between the force and displacement vectors.

Gravitational potential energy is the energy an object possesses due to its position relative to other objects. It is defined as the product of the object's mass, the acceleration due to gravity (9.8 m/s^2), and its height above the reference point.

Now let's calculate the values step by step:

(a) To find the work done on the basketball by its weight, we need to calculate the gravitational force acting on it. The weight of an object can be determined using the formula: Weight = mass x gravity, where gravity is the acceleration due to gravity.

Given that the mass of the basketball is 0.662 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate the weight:
Weight = 0.662 kg x 9.8 m/s^2 = 6.48 N

The displacement of the basketball is the distance from the window to the hands of the person, which is the height of the window minus the height of the hands:
Displacement = (6.32 m - 1.82 m) = 4.5 m

The angle between the force of gravity and the displacement is 180 degrees (cosine of 180 degrees is -1).

So, the work done on the basketball by its weight is given by:
Work = 6.48 N x 4.5 m x -1 = -29.16 Joules

Therefore, the work done on the basketball by its weight is -29.16 Joules.

(b) The gravitational potential energy of the basketball relative to the ground when it is released can be calculated using the formula:
Potential Energy = mass x gravity x height

Given that the mass is 0.662 kg, the height from the ground is 6.32 m, and the acceleration due to gravity is 9.8 m/s^2, we can calculate:
Potential Energy = 0.662 kg x 9.8 m/s^2 x 6.32 m = 40.54 Joules

Therefore, the gravitational potential energy of the basketball when it is released is 40.54 Joules.

(c) The gravitational potential energy of the basketball relative to the ground when it is caught can be calculated using the same formula, but with the height of the person's hands (1.82 m):
Potential Energy = 0.662 kg x 9.8 m/s^2 x 1.82 m = 11.68 Joules

Therefore, the gravitational potential energy of the basketball when it is caught is 11.68 Joules.

(d) The change in gravitational potential energy (PEf - PE0) can be calculated by subtracting the initial potential energy (when released) from the final potential energy (when caught):
Change in Potential Energy = Final Potential Energy - Initial Potential Energy
Change in Potential Energy = 11.68 Joules - 40.54 Joules = -28.86 Joules

Therefore, the change in the basketball's gravitational potential energy is -28.86 Joules.