Question

analyze the energy of the ball alone and of the ball-earth system throughout this process. then complete the table. assume air resistance is small enough to ignore. event kinetic energy of ball total energy of ball-earth system the ball swings down after the cable is released the ball smashes into the wall increases decreases stays the same

Answer:

For the event "the ball swings down after the cable is released":

Kinetic energy of the ball:

• As the ball swings down, its height decreases (so gravitational potential energy decreases), and since we ignore air resistance, by the conservation of mechanical energy (for the ball - Earth system, but for the ball alone, the force of gravity is an external force doing work on it), the work done by gravity increases the ball's speed. Kinetic energy is given by $KE=\frac{1}{2}mv^{2}$, so as speed $v$ increases, kinetic energy increases. So the kinetic energy of the ball increases.

Total energy of ball - Earth system:

• In the ball - Earth system, the only forces involved are internal (gravity is an internal force between the ball and the Earth). Since we ignore air resistance (no external non - conservative forces doing work), the total mechanical energy (kinetic + potential) of the system is conserved. So the total energy of the ball - Earth system stays the same.

For the event "the ball smashes into the wall":

Kinetic energy of the ball:

• When the ball smashes into the wall, the wall exerts a force on the ball. This force does negative work on the ball (opposing the ball's motion), which causes the ball's speed to decrease. Since $KE = \frac{1}{2}mv^{2}$, as speed $v$ decreases, the kinetic energy of the ball decreases.

Total energy of ball - Earth system:

• When the ball hits the wall, some of the ball's kinetic energy is transferred to the wall (as heat, sound, or work done to deform the wall or the ball). However, if we consider the ball - Earth system, the collision with the wall involves an external force (the normal force from the wall on the ball). But the question is about the total energy of the ball - Earth system. The energy lost by the ball (decrease in kinetic energy) is transferred to the wall (and its surroundings), but the ball - Earth system's energy: the potential energy of the ball - Earth system doesn't change much in the short time of the collision, and the kinetic energy of the ball decreases. But actually, in a more accurate sense, when the ball collides with the wall, the total mechanical energy of the ball - Earth system is not conserved because of the external force from the wall. But if we consider all forms of energy (including the energy transferred to the wall and the thermal energy, etc.), the total energy of the universe is conserved, but for the ball - Earth system, the total energy (mechanical + energy transferred out) - but in the context of this problem (probably considering mechanical energy initially), when the ball hits the wall, the kinetic energy of the ball decreases, and the total mechanical energy of the ball - Earth system also decreases (because energy is transferred out of the system to the wall and its surroundings). But maybe in the problem's context, for the ball - Earth system during the collision, the total energy (if we consider that the collision is inelastic and energy is dissipated) decreases. But let's re - evaluate:

Wait, the first event: "the ball swings down after the cable is released"

• Kinetic energy of ball: increases (as speed increases when falling down)
• Total energy of ball - Earth system: stays the same (conservation of mechanical energy, since only gravity acts, air resistance is ignored)

Second event: "the ball smashes into the wall"

• Kinetic energy of ball: decreases (because the wall applies a force that slows the ball down)
• Total energy of ball - Earth system: decreases (because the kinetic energy of the ball is converted into other forms of energy like heat, sound, and deformation energy, which are outside the ball - Earth system's mechanical energy. So the total mechanical energy of the ball - Earth system decreases, and if we consider all energy, but in the context of the problem's table, likely the answer for total energy of ball - Earth system when the ball smashes into the wall is decreases.

Filling the table:

Event	Kinetic energy of ball	Total energy of ball - Earth system
the ball smashes into the wall	decreases	decreases

(Note: The options for each dropdown are "increases", "decreases", "stays the same". So for the first event's kinetic energy of ball: select "increases"; total energy of ball - Earth system: select "stays the same". For the second event's kinetic energy of ball: select "decreases"; total energy of ball - Earth system: select "decreases".)