A 0.582-kg basketball is dropped out of a window that is 5.98 m above the ground. The ball is caught by a person whose hands are 1.58 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?
To find the answers to these questions, we'll need to use some equations and principles from classical mechanics.
(a) To calculate the work done on the ball by its weight, we'll use the equation:
Work = Force x Distance x cos(theta)
In this case, the force is the weight of the ball (mg), the distance is the height it is dropped (5.98 m), and theta is the angle between the force and the direction of motion (in this case, the force and motion are both vertically downward so cos(theta) = 1).
The weight of the ball can be calculated using the equation:
Weight = mass x acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
So, first we'll calculate the weight of the ball:
Weight = 0.582 kg x 9.8 m/s^2 = 5.70276 N
Then we'll calculate the work done:
Work = 5.70276 N x 5.98 m x 1 = 34.18 J
Therefore, the work done on the ball by its weight is 34.18 Joules.
(b) The gravitational potential energy (PE) of the basketball when it is released can be calculated using the equation:
PE = mass x gravitational field strength x height
The gravitational field strength on Earth is approximately equal to the acceleration due to gravity which is 9.8 m/s^2.
PE = 0.582 kg x 9.8 m/s^2 x 5.98 m = 34.44116 J
Therefore, the gravitational potential energy of the basketball when it is released is approximately 34.44 Joules.
(c) Similarly, the gravitational potential energy of the basketball when it is caught can be calculated using the same equation with the height being the difference between the height it is caught and the ground.
Height = 1.58 m
PE = 0.582 kg x 9.8 m/s^2 x 1.58 m = 9.169976 J
Therefore, the gravitational potential energy of the basketball when it is caught is approximately 9.17 Joules.
(d) Finally, we can calculate the change in the ball's gravitational potential energy:
Change in PE (PEf - PE0) = PEf - PE0 = 9.17 J - 34.44 J = -25.27 J
The change in the ball's gravitational potential energy is approximately -25.27 Joules.