Question

Predict \& Explain When a ball of mass \(m\) is dropped from rest from a height \(h\), its kinetic energy just before landing is \(K E\). Now, suppose a second ball of mass \(4 m\) is dropped from rest from a height \(h / 4\). (a) Just before ball 2 lands, is its kinetic energy \(4 K E, 2 K E, K E, K E / 2\), or \(K E / 4\) ? (b) Choose the best explanation from among the following: A. The two balls have the same initial energy. B. The more massive ball will have the greater kinetic energy. C. The lower drop height results in a reduced kinetic energy.

Short answer

(a) KE; (b) Explanation A.

Step-by-step solution

Understand the Concepts

When a ball is dropped from a height, its gravitational potential energy at the beginning is converted into kinetic energy just before it hits the ground. The potential energy is given by the formula \( PE = mgh \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height. Thus, the kinetic energy just before impact is equal to the initial potential energy.

Calculate Kinetic Energy of Ball 1

For the first ball of mass \( m \) dropped from height \( h \), its potential energy is \( mgh \). This is fully converted into kinetic energy \( KE \) just before landing, so \( KE = mgh \).

Calculate Kinetic Energy of Ball 2

For the second ball of mass \( 4m \) dropped from height \( h/4 \), its initial potential energy is \( PE = 4m \cdot g \cdot \frac{h}{4} = mgh \). Thus, the kinetic energy just before it lands is also \( mgh \).

Compare Kinetic Energies of Both Balls

Both balls have a kinetic energy of \( mgh \) just before they land. Therefore, the kinetic energy of the second ball is equal to that of the first ball, which is \( KE \).

Determine the Correct Explanation

Based on our calculations, the two balls have the same kinetic energy just before landing because their initial potential energies were equal. The correct explanation is that the initial energy was the same for both balls, denying options B and C.

Key concepts

Gravitational Potential Energy

Gravitational potential energy is a form of energy that an object possesses due to its position in a gravitational field, commonly associated with height above the ground. The formula to calculate gravitational potential energy is \( PE = mgh \), where:

• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity (typically \( 9.8 \, \text{m/s}^2 \) on Earth),
• \( h \) is the height above the ground.

When an object is held at a certain height, it stores potential energy. This energy is a direct product of the object's mass, the gravitational pull of the Earth, and how high it is situated. As seen in the exercise, when either of the balls is dropped, its gravitational potential energy is transformed into kinetic energy just before impact.

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion. The formula used to express this form of energy is \( KE = \frac{1}{2}mv^2 \), where:

• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

In the context of the exercise, when the balls are released and begin to fall, the gravitational potential energy begins converting into kinetic energy. Just before the balls hit the ground, all of the initial potential energy they possessed has become kinetic energy.
An important point to note is that regardless of the ball's mass or initial height (given similar initial potential energies), the kinetic energy just before striking the ground depends strictly on the conversion from potential energy, which results in the same kinetic energy for the two balls described in the exercise.

Mass and Height Relationship

The relationship between mass and height plays a crucial role in determining gravitational potential energy. Increasing either the mass or the height increases the potential energy:

• Doubling the mass while keeping the height constant will double the potential energy.
• Doubling the height while maintaining the same mass will also double the potential energy.

In the exercise discussed, the first ball's energy depends on a certain height \( h \), and a mass \( m \). By contrast, the second ball has four times the mass but is dropped from a quarter of the height. Despite these differences, both scenarios yield the same potential energy \( mgh \), thus leading to the same kinetic energy just before impact.
This exercise illustrates that even with different mass and height parameters, the potential energies can be made equivalent by proportionally adjusting mass and height, leading to equal kinetic energies just before impact.