A ball of mass m falls from height h,to height hf near the surface of the Earth, as shown at right. When the ball passes it has a speed vf. Ignore air resistance, and assume that any changes in kinetic energy of the Earth are negligible.

In this problem, you will analyze this situation by considering two different systems: system BE (the ball and the Earth) and system B (the ball alone).

a. Consider system BE, consisting of the ball and the Earth.

i. As the ball falls from h{ to hf, does the total energy of system BE increase, decrease, or stay the same?) Explain by considering the net work by external forces on system BE.

ii. Consider the following discussion:

Student 1: "The changes to system BE are all internal, so its gravitational potential energy doesn't change. Since the kinetic energy of the ball increases, the total energy of system BE increases."

Student 2: "I think that the change in potential energy of the system is zero because the distance the ball moves is small compared to the size of the earth. The change in total energy of system BE is equal to the change in kinetic energy of the ball."

Identify the incorrect statements made by both students. Explain.

iii. Write an expression for each of the following quantities in terms of the given variables and any physical constants. If any of these are zero, state so explicitly.

• the change in kinetic energy of system BE
• the change in potential energy of system BE
• the net external work by external forces on system BE
iv. Write an equation that relates the expressions above.

v. Solve your equation from part iv for the final speed of the ball. Is your result consistent with what you would obtain using kinematics and your knowledge of the acceleration due to gravity near the Earth's surface? If not, resolve the inconsistency.

b. Consider the system of the ball only, system B.

i. As the ball falls from h{ to hf, does the total energy of system B increase, decrease, or stay the same? Explain by considering the net external work on system B.

ii. Write an expression for each of the following quantities in terms of the given variables and any physical constants.

• the net external work on system B
• the change in kinetic energy of system B
• the change in potential energy of system B

iii. Consider the following true statement about potential energy:

"Changes in potential energy are associated with changes in shape of a system, or changes in relative positions of the objects that make up the system. A system consisting of a single object that undergoes no change in shape or other internal changes does not have a change in potential energy."

Explain how your answer to the third bullet of part b.ii is consistent with this statement. If it is not consistent, how could you change it to make it consistent?

iv. Write an equation that relates the three expressions from part b.ii.

v. Solve the equation from part b.iv for the final speed of the ball.

c. Check that your answer to part b.v is consistent with your answer to part a.v. If not, resolve the inconsistency.

d. In summary, based on work and energy considerations:

i. Can the Earth affect the total energy of system BE (i.e., a system that includes the Earth)? Explain.

ii. Can the Earth affect the total energy of system B (i.e., a system that does not include the Earth)? Explain.