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Predict/Explain Ball 1 is thrown to the ground with an initial downward speed; ball 2 is dropped to the ground from rest. Assuming the balls have the same mass and are released from the same height, is the change in gravitational potential energy of ball 1 greater than, less than, or equal to the change in gravitational potential energy of ball \(2 ?\) (b) Choose the best explanation from among the following: I. Ball 1 has the greater total energy, and therefore more energy can go into gravitational potential energy. II. The gravitational potential energy depends only on the mass of the ball and the drop height. III. All of the initial energy of ball 2 is gravitational potential energy.

Step by step solution

01

Understanding Potential Energy Change

The change in gravitational potential energy depends solely on the initial and final positions of an object in a gravitational field. It is calculated as \( \Delta U = mgh_i - mgh_f \), where \( m \) is the mass, \( g \) is the gravitational acceleration, and \( h \) is the height. Since both balls are released from the same height and fall to the same final height, their changes in potential energy will be equal.

02

Analyzing Initial Conditions

Ball 1 is thrown downwards with an initial speed, while Ball 2 is dropped from rest. This influences the kinetic energy at the moment of release but not the potential energy. The initial potential energy is the same for both balls as they start from the same height.

03

Evaluating the Explanation Options

Option I is incorrect as it talks about total energy, not just potential energy. Option III is true but does not address potential energy change. Option II correctly states that potential energy change depends only on mass and height, thus making Option II the correct answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Physics Problem Solving

When faced with physics problems, it is essential to break down the situation step by step. This structured approach helps in comprehending the details and applying the correct principles to find a solution. For instance, with the problem involving the two balls, it begins by clearly identifying what is being asked: the change in gravitational potential energy. This requires understanding the relevant physics formula and parameters involved, such as mass, gravitational force, and height.
Breaking down the problem:

• Define the variables: mass of balls, initial and final heights.
• Understand the concepts of potential and kinetic energy.
• Assess how initial conditions affect these energies.

This logical method ensures that one can systematically tackle each part to arrive at the correct conclusion.

Energy Conservation

The principle of energy conservation plays a crucial role in solving physics problems. It states that in a closed system, the total energy remains constant. This is pivotal when analyzing how different forms of energy interact and transform into one another. In this context, both kinetic and gravitational potential energy are involved.
For the two balls:

• The gravitational potential energy depends solely on height and not on how the balls move initially.
• Ball 1's initial velocity influences its kinetic energy, but not its potential energy.
• Energy transformations occur as potential energy converts to kinetic energy during the fall.

Understanding that potential energy changes depend only on height sets a solid foundation for applying the conservation of energy in such problems.

Initial Conditions Analysis

Analyzing initial conditions is crucial to understand their effect on the energies involved. For the two balls, both are released from the same height, ensuring that their gravitational potential energy at the start is equal. However, their kinetic energy differs due to Ball 1 having an initial downward velocity.
Initial condition factors:

• Both balls start from identical heights, ensuring equal initial potential energy.
• The distinct initial velocities influence the kinetic energy present at the beginning.
• Despite these differences, the drop height determines the potential energy change, not the initial motion.

By focusing on these initial conditions, we reinforce that for gravitational potential energy changes, only the mass and amount of height fallen matters.